Can anyone give a combinatorial proof of the identity ${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+...+(n-m+1){m \choose m}={n+2 \choose m+2}$

HINT: You want to count the subsets of $\{0,1,\ldots,n+1\}$ that have $m+2$ elements. Clearly there are $\binom{n+2}{m+2}$ of them. However, we can also count them in batches according to the second-smallest number in the set.

How many $(m+2)$-element subsets of $\{0,1,\ldots,n+1\}$ have $k$ as their second-smallest element?