[Comment number 2, describing what this theorem says for people with patience but not necessarily much mathematical expertise. Read comment number 1 first. There isn't a number 3.]
OK, time for ingredient number 2 which is the idea of a manifold. Roughly, an n-dimensional manifold is a geometrical object which, whenever you look at it closely enough, looks like "ordinary" n-dimensional "Euclidean" space. Once again, let's begin in dimension 2. "Ordinary" 2-dimensional space is something mathematicians write as R^2: jus as Z^2 consisted of pairs of integers, R^2 consists of pairs of real numbers. (Real numbers are what you probably think of just as "numbers".) We can think of this as an infinite plane; the two numbers are coordinates. Now, for a fairly typical example of a 2-dimensional manifold, consider a sphere -- the surface of the earth, say, if you smooth that out a bit. Any small portion of it looks just like a small portion of the plane, which is why there is a Flat Earth Society. But the whole thing has a different structure -- e.g., the sphere is finite in extent in a way the plane isn't. Another example is the surface of a ring do(ugh)nut. This is also finite in extent, but it turns out to be genuinely different from the sphere, and there's a whole lot of interesting mathematics around that, which I am going to ignore here.
Now, just as we considered those very special transformations acting on Z^2 before, we are going to consider transformations acting on a manifold: things that pair up each point on the manifold "before" the transformation with some point "after" it. For instance, suppose we take the (idealized) surface of the earth; one example of a transformation would be what you get by rotating the earth about its axis through 10 degrees so that every point moves west (and e.g. Kansas City lands more or less on top of Denver). But our transformations can be more complicated: imagine our sphere to be a thin sheet of very flexible rubber forced somehow to remain on the surface of a globe; we can then push things around however we like provided the sheet never tears or overlaps itself. These things (with some technical restrictions I won't go into) are called diffeomorphisms. And, though it's harder to visualize, we can do much the same thing with any manifold of any dimension. If M is our manifold then we call the set of all its diffeomorphisms Diff(M). And, just like SL(n,Z), this thing is a group: we can compose two diffeomorphisms to get another diffeomorphisms, and because we insisted on no tearing or double-covering every diffeomorphism can be undone by another diffeomorphism.
All right, nearly there. Ingredient number 3 is the idea of a group homomorphism. Suppose we have two groups; call them G and H. Suppose that for each element of G we somehow pick out an element of H. We'll write f(g)=h to mean that element g in G yields element h in G; "f" is the name we're giving to our correspondence between G and H. Unlike the transformations considered above, we aren't going to insist on any sort of invertibility; f might map lots of different g to the same h, and there might be some h that aren't the "image" of any g. Here's a simple example: consider SL(2,Z) again, and consider a manifold M that's just ordinary 2-dimensional space, what we called R^2. Everything in SL(2,Z), remember, maps (x,y) to (ax+by,cx+dy) -- and we can do that just as well when x,y are arbitrary real numbers as when they are integers, and the resulting thing is in fact a diffeomorphism. So everything in SL(2,Z) gives rise to a thing in Diff(R^2).
In this case, because these are "the same" transformation in some sense, composition of things in SL(2,Z) matches up nicely with composition of things in Diff(R^2). This sort of matching-up-nicely can happen even in less straightforward cases. If f(g1 compose g2) always equals f(g1) compose f(g2) then we call f a group homomorphism between SL(n,Z) and Diff(M). The specific transformations in SL(n,Z) and in Diff(M) needn't have anything much to do with one another -- but the relationships between them need to have compatible structures, in some sense.
So, we saw above that you can embed a copy of SL(2,Z) inside Diff(R^2). More generally, there's a copy of SL(n,Z) inside Diff(R^n). What these guys have done is to put some limits on correspondences of this kind between SL(n,Z) and Diff(R^m) where m is smaller than n: the idea is that SL(n,Z) is an n-dimensional thing and that you can't squash it into something of much lower dimension without destroying its structure.
Unfortunately there's one more bit of technical detail needed before stating their result. Ingredient number 4 is the idea of a finite-index subgroup. Zimmer's conjecture restricts not only group homomorphisms from SL(n,Z) itself, but also from certain smaller things that in some sense contain most of the structure of SL(n,Z). So, suppose you have some subset S of SL(n,Z) which is also a group: composites and inverses of things in S are always in S themselves. Then it turns out that SL(n,Z) can be partitioned into "copies" of S. One is S itself; if x is anything that isn't in S, then the compositions "x compose s", as s runs over everything in S, form another of these copies, and x itself is in that copy. Any two of these copies are either identical or disjoint, it turns out; so the whole of SL(n,Z) is made up of a bunch of these things. If there are only finitely many of these disjoint copies, we say that S has "finite index"; so e.g., maybe there are 12 of them, meaning that S is in some sense 1/12 as big as all of SL(2,Z). (Just to be clear, SL(n,Z) is infinite.)
We can finally state the theorem properly: If S is a finite-index subgroup of SL(n,Z), and f : S -> Diff(M) is a group homomorphism, and the dimension of M is m where m < n-1, then the image of f -- the set of things in Diff(M) corresponding to things in S -- is finite.
That is: you can't fit something with the same structure as "most of" SL(n,Z) into the diffeomorphisms of something with dimension < n-1, unless you collapse that structure "almost completely".
Sorry, just a posting note: If you want your comments to display in the right order, it would make more sense to have your second comment be a reply to your first, rather than having them as sibling comments.
OK, time for ingredient number 2 which is the idea of a manifold. Roughly, an n-dimensional manifold is a geometrical object which, whenever you look at it closely enough, looks like "ordinary" n-dimensional "Euclidean" space. Once again, let's begin in dimension 2. "Ordinary" 2-dimensional space is something mathematicians write as R^2: jus as Z^2 consisted of pairs of integers, R^2 consists of pairs of real numbers. (Real numbers are what you probably think of just as "numbers".) We can think of this as an infinite plane; the two numbers are coordinates. Now, for a fairly typical example of a 2-dimensional manifold, consider a sphere -- the surface of the earth, say, if you smooth that out a bit. Any small portion of it looks just like a small portion of the plane, which is why there is a Flat Earth Society. But the whole thing has a different structure -- e.g., the sphere is finite in extent in a way the plane isn't. Another example is the surface of a ring do(ugh)nut. This is also finite in extent, but it turns out to be genuinely different from the sphere, and there's a whole lot of interesting mathematics around that, which I am going to ignore here.
Now, just as we considered those very special transformations acting on Z^2 before, we are going to consider transformations acting on a manifold: things that pair up each point on the manifold "before" the transformation with some point "after" it. For instance, suppose we take the (idealized) surface of the earth; one example of a transformation would be what you get by rotating the earth about its axis through 10 degrees so that every point moves west (and e.g. Kansas City lands more or less on top of Denver). But our transformations can be more complicated: imagine our sphere to be a thin sheet of very flexible rubber forced somehow to remain on the surface of a globe; we can then push things around however we like provided the sheet never tears or overlaps itself. These things (with some technical restrictions I won't go into) are called diffeomorphisms. And, though it's harder to visualize, we can do much the same thing with any manifold of any dimension. If M is our manifold then we call the set of all its diffeomorphisms Diff(M). And, just like SL(n,Z), this thing is a group: we can compose two diffeomorphisms to get another diffeomorphisms, and because we insisted on no tearing or double-covering every diffeomorphism can be undone by another diffeomorphism.
All right, nearly there. Ingredient number 3 is the idea of a group homomorphism. Suppose we have two groups; call them G and H. Suppose that for each element of G we somehow pick out an element of H. We'll write f(g)=h to mean that element g in G yields element h in G; "f" is the name we're giving to our correspondence between G and H. Unlike the transformations considered above, we aren't going to insist on any sort of invertibility; f might map lots of different g to the same h, and there might be some h that aren't the "image" of any g. Here's a simple example: consider SL(2,Z) again, and consider a manifold M that's just ordinary 2-dimensional space, what we called R^2. Everything in SL(2,Z), remember, maps (x,y) to (ax+by,cx+dy) -- and we can do that just as well when x,y are arbitrary real numbers as when they are integers, and the resulting thing is in fact a diffeomorphism. So everything in SL(2,Z) gives rise to a thing in Diff(R^2).
In this case, because these are "the same" transformation in some sense, composition of things in SL(2,Z) matches up nicely with composition of things in Diff(R^2). This sort of matching-up-nicely can happen even in less straightforward cases. If f(g1 compose g2) always equals f(g1) compose f(g2) then we call f a group homomorphism between SL(n,Z) and Diff(M). The specific transformations in SL(n,Z) and in Diff(M) needn't have anything much to do with one another -- but the relationships between them need to have compatible structures, in some sense.
So, we saw above that you can embed a copy of SL(2,Z) inside Diff(R^2). More generally, there's a copy of SL(n,Z) inside Diff(R^n). What these guys have done is to put some limits on correspondences of this kind between SL(n,Z) and Diff(R^m) where m is smaller than n: the idea is that SL(n,Z) is an n-dimensional thing and that you can't squash it into something of much lower dimension without destroying its structure.
Unfortunately there's one more bit of technical detail needed before stating their result. Ingredient number 4 is the idea of a finite-index subgroup. Zimmer's conjecture restricts not only group homomorphisms from SL(n,Z) itself, but also from certain smaller things that in some sense contain most of the structure of SL(n,Z). So, suppose you have some subset S of SL(n,Z) which is also a group: composites and inverses of things in S are always in S themselves. Then it turns out that SL(n,Z) can be partitioned into "copies" of S. One is S itself; if x is anything that isn't in S, then the compositions "x compose s", as s runs over everything in S, form another of these copies, and x itself is in that copy. Any two of these copies are either identical or disjoint, it turns out; so the whole of SL(n,Z) is made up of a bunch of these things. If there are only finitely many of these disjoint copies, we say that S has "finite index"; so e.g., maybe there are 12 of them, meaning that S is in some sense 1/12 as big as all of SL(2,Z). (Just to be clear, SL(n,Z) is infinite.)
We can finally state the theorem properly: If S is a finite-index subgroup of SL(n,Z), and f : S -> Diff(M) is a group homomorphism, and the dimension of M is m where m < n-1, then the image of f -- the set of things in Diff(M) corresponding to things in S -- is finite.
That is: you can't fit something with the same structure as "most of" SL(n,Z) into the diffeomorphisms of something with dimension < n-1, unless you collapse that structure "almost completely".