What physically/chemically happens to a vegetable when it is cooked?
The main difference is the degradation of cell walls. The human digestive system does indeed struggle to break down the cellulose in cell walls, meaning that we are far more efficient in getting nutrients from vegetables when they are cooked.

That doesn't mean however that cooked veggies are always more nutritious than raw ones. There is actually quite a bit of debate on that and some people even eat raw foods exclusively.

Your question is actually very similar to this earlier question on this site. You might find some useful insights there.