It may be of help to think this in terms of the image instead of the rank: T maps the whole space V into W, but if S doesn't "cover" V then T can't "use" the elements that S missed.
if S : U-->V and T : V---> W are linear maps, and U,V and W are vector spaces over the same field K, how can we prove the following:
1) rank (TS) < or = rank (T)
2) rank (TS) < or = rank (S)
3) if U=V and S is nonsingular then Rank (TS) = Rank (T)
4) if V=W and T is nonsingular then Rank (TS) = Rank (S)