So Y is the intersection of all n-1 dimensional subspaces that contain U.

Choose a basis for subspace W. Extend that basis to a basis for V. Every n- 1 dimensional subspace, that contains W, is spanned by that basis, with one basis vector removed (not one of those in the basis for W, of course). It should be clear that the original basis for W spans the intersection of those.

It would be great if I could also get some explanations for more complex things abut why something works as it works (in order to understand it better). Or if someone has seen proof to this on some book, link would be great as well.

hint: to show that $\displaystyle Y \supset W$ we show that $\displaystyle v \in Y \backslash W$ leads to conflict