Problem:

Layman here. So I have never really quite understood this facet of human evolution, (or any other for that matter), in that, I understand the evolutionary process, but I get lost on the 'border' cases.
For example, we, as humans, evolved from monkeys, (to use the colloquial term, I am not a biologist by any measure).

Question: Doesn't this mean that at some, discrete point, there had to have been a human, whose parents were not?

If that is true,

Question: how does that work, in the sense that we now have species1 giving birth to species2.
Explain your answer.

Question : If not, then how exactly does this border case work?

The only other alternative I see, is that the borders are 'fuzzy', but then that necessarily means that the definition of a species is itself fuzzy, which I understand is not the case.