Why, conceptually, is it that $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$? Why, conceptually, is it that $$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}?$$ I know how to prove that this is true, but I don't understand conceptually why it makes sense.

Another simple way to look at it is Pascal's triangle.  This is simply a formulation of the rule that governs the triangle -- an entry is the sum of the two entries above it.  The $i$-th entry in the $n$-th row is the sum of the $i-1$-th entry in the $n-1$-th row and the $i$-th entry in the $n-1$-th row.  Hence,
$$\binom{n-1}{i-1}+\binom{n-1}{i}=\binom{n}{i}$$
