Actually, in real lenses, the ray going through the center of the lens does NOT go straight through. It refracts like every other light ray. On entering the lens from air, the light ray is bent. Upon exiting, the ray is bent back. Take a simple chunk of glass like a rectangular prism. Shoot a beam of light at an angle through it, and the beam coming out of it would be parallel to the original beam, but offset by a small amount, depending on the thickness of the glass.
When you first learn geometric optics, a lot of this might get glossed over and hence the trouble with determining WHY the simplified ray diagrams work. The thinner the block of glass, the less offset the two parallel rays are to each other. If you are dealing with lenses that have the same curvature on both sides - like most lenses in beginning physics, then you can imagine that you have a chunk of glass with parallel sides that this light is interacting with and the thickness of this chunk of glass is so small that the resulting offset is basically negligible.
So the ray through the center of a thin lens appears to go straight through, but not because it is not refracted as st-2 suggests.
What bigmama97 said is partially true. A ray incident at 90° to a surface does travel straight through. However, this doesn't help any with lenses, because the means that ray would be pointed at the center of curvature for only one side of the lens, the ray exiting would not end up at 90° to the other side, since the center of curvature of the opposite side is somewhere else.
Now, if you were using a glass sphere, then, and only then, the light ray directed through the center would go unaltered through both sides of the glass - but this is just like the rectangular prism mentioned earlier.