Quantum field theory is very difficult and we can't solve many problems. We can solve one in particular, free particles that do not interact. (By the way, this is like solving a simple harmonic osciallator.) Feynman diagrams use the solution to free particles as a basis - these are the lines in the diagram, like particles moving in time. Then, we add the interactions between the particles as a pertabative exapansion. This is what the parent comment refers to as the power series expansion. It is expanding in the powers of the interaction terms.
As mentioned in the video, it is not quite as simple as this however. If you do this, you will get infinity as a result of the calculation. That is not a very good perterbation theory. However the real discovery of physcists from this time was renormalization. They found out you can sum many contributions from different terms in the exapansion, corresponding to different indiviaul feynmann diagrams, and you can make these infinities cancel out. What is magic is that when this process is done you can write an effective theory for the interactions that looks exactly like the original theory you started with, and only the values of the coeffiecients are changed a little.
The magic involved here could be explained more, but in summary what it means is that the perterbation calculation you do is valid even thought it looks like it should be infinite. It also lets you make the simple analogy that the lines in the diagram are real and virtual particles rather than just the non-interacting approximation you starter with.
I left out some important details in the above explanation. So here is part 2...
Feynman discovered another very important item, the path integral formulation of physics. This was important for his derivation of Feynman diagrams and also it is a good conceptual tool.
Think of basic quantum mechanics and firing an electron through a double slit at a screen. In quantum mechanics the eletron does not have a single trajectory from the gun to the screen. Rather, it takes all the trajectories in parallel, like a wave. We can add the contribution as if the particle went over each possible trajectory and this is the same as treating the electron as if its position was given by a propogating wave. This sum over possible histories is the interpretation of Feynmans path integrals. And it is a nice way to think about quantum mechanics - multiple things are happening in parallel.
Taking the example in the video of two electrons scattering off each other, each Feynman diagram represents a possible history for the two particles, including their trajectory and any interactions between them. These interactdions are drawn as a connecting line, which is a "virtual particle" being exchanged.
More specifically, the diagram doesn't represent a single history, rather it represents all histories that have a ceratain topology, meaning here for example one photon is exchanged between the electrons. (There is an integral done to add up all the different ways this can happen.) To do the full calculation, there are many diagrams that must be included. As it is a perterbation theory, you can choose to get more accurate by include more diagrams. The expansion parameter is basically a vertex on the diagram. The more vertices you include, the more accurate you will be (assuming you include all diagrams with that number of vertices).
So basically the feynman diagram is a bookkeeping mechanism to account for all possible histories of the particles in the interaction (electon scattering here). We sum up the contribution from all these histories to find out the quantum amplitude for this scattering scenario. This is exactly analogous to adding the contribution from different paths to fine the amplitude (~probability) for our electron in the double slit experiment hitting a particular location on the screen.
To get to this intuitive result mathematically, the perturbation expansion and renormalization mentioned above are both involved.
Quantum field theory is very difficult and we can't solve many problems. We can solve one in particular, free particles that do not interact. (By the way, this is like solving a simple harmonic osciallator.) Feynman diagrams use the solution to free particles as a basis - these are the lines in the diagram, like particles moving in time. Then, we add the interactions between the particles as a pertabative exapansion. This is what the parent comment refers to as the power series expansion. It is expanding in the powers of the interaction terms.
As mentioned in the video, it is not quite as simple as this however. If you do this, you will get infinity as a result of the calculation. That is not a very good perterbation theory. However the real discovery of physcists from this time was renormalization. They found out you can sum many contributions from different terms in the exapansion, corresponding to different indiviaul feynmann diagrams, and you can make these infinities cancel out. What is magic is that when this process is done you can write an effective theory for the interactions that looks exactly like the original theory you started with, and only the values of the coeffiecients are changed a little.
The magic involved here could be explained more, but in summary what it means is that the perterbation calculation you do is valid even thought it looks like it should be infinite. It also lets you make the simple analogy that the lines in the diagram are real and virtual particles rather than just the non-interacting approximation you starter with.