A lens takes all the light rays coming from a certain direction and bends them so they all end up in the same place. In the case of your eye, they need to end up on the retina. When your pupil is larger, you have to bend the rays coming in at the edge more than the rays going through the center of your pupil. Terrible vision (usually) happens because the lens of your eye isn't bending the light rays by the right amount. Let's say your lens only bends them half as much as it should. Light going through the center of your pupil isn't affected because it they go straight through. Rays near the center get bent by the wrong amount, but because they weren't getting bent very much still end up in nearly where they should have. But rays that come through the edge of your pupil need to get bent by a lot, so when your lens isn't working right, they end up a long ways from where they should. That makes your vision fuzzy, because the light from say a single light bulb ends up spread all over your retina. If you block the outer parts of your pupil, though, those rays that ended up a longs ways off and made your vision fuzzy get blocked, so the light that does make it through ends up where it is supposed to.

The limiting case of this is, as people have mentioned, a pinhole camera. The pinhole doesn't bend the light at all, so what you end up with is a picture that has been smeared out by the size of the pinhole. The smaller the pinhole, the sharper (and fainter) the image. The eye is better than a pinhole because it's at least trying to focus, but it is the same basic idea.