Why, conceptually, is it that $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$? [duplicate]

Because the number of ways that we can make a team of a certain size is equal to the number of ways that we can make a team of the same size without Fred, plus the number of ways we can make a team of size one smaller, plus Fred.

The number of the combination that we choose $r$ people from $n$ people is the sum of the number of the combination that a person named $A$ is included and the number of the combination that $A$ is not included.

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.