How is the Combinatorial proof derived?

Question

Could somebody explain how this statement is derived?

$${m+1\choose{k+1}} = \sum_{i=0}^{m-k} {k+i\choose{k}}$$

This is similar to

$\binom{m+1}{k+1}=\sum_{i=0}^m \binom{i}{k}$

but I am stuck as to how this has evolved into my original post.