Since you mentioned that you were having a hard time visualising this and I always seem to find myself visualising it whenever I have the need to write it down here is what goes through my mind as the pen is moving across the paper:

We are placing $r$ identical balls in $n$ boxes that are in a straight line, so ${ n \choose r}$ ways to do this, now either the last box is empty, that's ${n-1 \choose r}$ ways, or the last box is full, that's ${ n-1 \choose r-1}$ ways. QED

This is equivalent to Sivaram's answer but does away with the colours, which for the purposes of visualising is probably slightly easier.

Since you mentioned that you were having a hard time visualising this and I always seem to find myself visualising it whenever I have the need to write it down here is what goes through my mind as the pen is moving across the paper:

We are placing $r$ identical balls in $n$ boxes (at most one in each) that are in a straight line, so ${ n \choose r}$ ways to do this, now either the last box is empty, that's ${n-1 \choose r}$ ways, or the last box is full, that's ${ n-1 \choose r-1}$ ways. QED

This is equivalent to Sivaram's answer but does away with the colours, which for the purposes of visualising is probably slightly easier.