Assume f>r. The RHS can be understood as the number of ways to choose r+1 items from f+1 items. Imagine f+1 green balls lined up in front of you. Imagine that the r+1 items will be colored orange. There must be a rightmost orange ball. Suppose it's in the f+1 position. The other r balls must be chosen from f green balls. Now suppose the rightmost orange ball is in the f position. The other r balls must be chosen from f-1 positions. Can you see that these are the LHS terms, from rightmost to leftmost? Do you see why the sum ends at r choose r?