When you add vector spaces, you get a new vector space containing all possible sums of vectors within each space. As a concrete example, if U was the subspace of R^(3) of all vectors of the form (x, 0, 0) and W is the subspace of all vectors of the form (0, y, 0), then U + W would be the set of all vectors of the form (x, 0, 0) + (0, y, 0) = (x, y, 0). That means that every vector in U + W can be written as a vector in U plus a vector in W.