Since left and right and up and down are both the same, it's essentially as though you're looking through the back of the object. Imagine if you made a sign clear and looked through the back. Even though no reversal has taken place, the text looks backward. This is why things appear reversed.
If you are confused about how left and right are not actually switched, I would say the easiest way is to think in absolute terms, not relative. That is, don't think about left and right, think about east and west. Face north and paint your east (right) hand red. Now when you look at a mirror (you're still facing north), your east hand is red, but the reflection's east hand (think in real world terms here) is also red. That's because the reflection's horizontal aspects have not been reversed.
If you are confused about how left and right are not actually switched, I would say the easiest way is to think in absolute terms, not relative. That is, don't think about left and right, think about east and west. Face north and paint your east (right) hand red. Now when you look at a mirror (you're still facing north), your east hand is red, but the reflection's east hand (think in real world terms here) is also red. That's because the reflection's horizontal aspects have not been reversed.
Richard Feynman enjoys discussing this: http://www.youtube.com/watch?v=msN87y-iEx0&gl=CA