could someone please give me some hints about how to prove this?
Suppose that $\displaystyle a,~b,~\&~c$ are positive integers such that $\displaystyle a>b$.
Further, suppose that we have$\displaystyle a $ blue balls and $\displaystyle c+b $ red balls.
Now we put the balls all together in a box from which we randomly choose $\displaystyle b$ balls.
Surely you can see that we can choose from $\displaystyle 0\text{ to }b$ blue balls.
That is all that summation says.