This is a simple idea.
If v+W is a subspace then
If you studied group theory....
of a subgroup or and .
If and only if .
The reason is as follows, the relation that defines cosets is an equivalence relation, hence it divides cosets into disjoint sets, and the set that contains is . Thus, because of disjointness we have that if then .
Now by definition a vector space is a group (abelian) thus the same discussion in the preceding paragraph applies to a group as well.