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.

Sure.  You're dividing into 2 cases.  On the one hand you're saying I'm stuck choosing this element over here.  So now I have r-1 more choices to make out of n-1 things.  In the other case you're refusing that element.  Now you've eliminated a choice, but still must pick r elements.  These two cases are exhaustive and exclusive.
