The RHS is equal to the number of ways of selecting $r+1$ out of $n+1$ objects. If I sum the number of selections when the $(r+1)$th (i.e. the last) selection is the at the $(x+1)$th position, I get the RHS.
I verified it with some values but couldn't prove it mathematically.
The usual notation is $\sum_{x=r}^n\binom{x}{r}=\binom{n+1}{r+1}$. Since $\binom{x}{r}=\binom{x+1}{r+1}-\binom{x}{r+1}$, the sum telescopes to $\binom{n+1}{r+1}-\binom{r}{r+1}$. But the last term is, of course, $0$.