I usually think of a linear transformation as squeezing its domain down to its range (or image, if you prefer) while squeezing its nullspace down to zero. The theorem tells you the two amounts of squeezing are the same; the squeezing of the domain is the dimension of the domain minus the dimension of the range, and the squeezing of the nullspace is the dimension of the nullspace minus the dimension of the zero-space (which is zero).
EDIT: Maybe a numerical example will make my visualization clearer. Suppose the domain has dimension 17, and the range has dimension 12. Where did those extra 5 dimensions in the domain go? They must have gone to zero. So the dimension of the nullspace --- that's the 5 --- plus the dimension of the range --- that's the 12 --- equals the dimension of the domain --- the 17. The 17 gets squeezed down to 12 by virtue of the 5 getting squeezed down to zero.
This doesn't remotely resemble a proof, it's for motivation purposes only (but I thought that's what was being requested).