What is the combinatorial meaning of $\sum\limits_{R = 0}^{N}\binom Rr\binom{N-R}{n-r} = \binom{N+1}{n+1}$? What is the combinatorial meaning of $$\sum_{R = 0}^{N}\binom Rr\binom{N-R}{n-r} = \binom{N+1}{n+1} \quad?$$ If the summing is performed with respect to $r$, i.e. $$\sum_{r = 0}^{n}\binom Rr\binom{N-R}{n-r} = \binom Nn,$$ then it has clear combinatorial meaning: the number of combinations by taking $n$ balls from an urn containing $R$ red balls and $N-R$ white balls.

This is an adaptation of this answer to another question.
You are asking about an identity that reads, for $k,m,n\in\mathbf N$:
$$
  \sum_{i=0}^k\binom im\binom{k-i}n=\binom{k+1}{m+n+1}.
$$
It is a variation of the Vandermonde identity, which reads, for $k,l,m\in\mathbf N$:
$$
  \sum_{i=0}^m\binom ki\binom l{m-i}=\binom{k+l}m,
$$
and which is combinatorially explained by counting the possibilities to choose a team of $m$ children among $k$ girls and $l$ boys. 
Here is a cominatorial explaination for the variation asked about. Choose a subset of $m+n+1$ numbers $a_j$ from the $k+1$-set $\{0,\ldots,k\}$, and arrange them in increasing order $0\leq a_0<\cdots<a_{m+n}\leq k$. Put $i=a_m$, then there are $\binom im$ choices left for $a_0,\ldots,a_{m-1}$, and $\binom{k-i}n$ choices for $a_{m+1},\ldots,a_{m+n}$.
