This ranges from "It doesn't." to "It's Magick!" Both of these, of course, are really non-explanations.
My favorite theory is that it works because it's really a focus.
According to this theory, all of us really know all kinds of things about what's going on all around us, but we keep this knowledge at a subconscious level. At that level, too, we are able to communicate with the people around us, so we can absorb the knowledge that they have.
The only trick is to get the knowledge from the subconscious to the conscious. We do that in dreams, and in flashes of insight, and in vague feelings. But we have often been taught to ignore our insight and feelings, and many of us forget our dreams.
The Tarot, though, is physical. There it is. So we are more likely to be able to pay attention to it. When a reader looks at a tarot card, she can allow the feelings and insight to come out. And because there is a card to point to, it's easier to express them, and easier for the seeker to listen.
All of us can be readers, just as all of us have dreams. The trick is just to relax and let it come.
The correct cards show up, I think, because those are the cards we really know we need. The universe is very accommodating about giving us what we need (not to be confused with what we think we need, or what we want!)
So the more in touch you become with your feelings and intuitions, the less you fight your subconscious mind, the better your readings will be. Try it!