" When you were born did your uncle(s) and cousins go *poof* into nothing-ness?"
No they didn't.
But I didn't evolve into a different species either.
This is another common misunderstanding. Every creature is always the same species as its parents. It is quite hard to explain, but one possible analogy is a long bar that slowly changes from black to white along its length. Black is one species and white is another, and the bar is thousands of generations long. No-one can point at the position on the bar where black changed to white, it is just merging greys all the way along.
No they didn't. They're still H. sapien I suppose.
But they are different H. sapien(s) with subtle differences in hair, eyes, blood type, and a whole host of other distinctions. And their descendants will be different from your descendants, even more so.
And so on, etc.
Eventually you get to a point where a reasonable person will acknowledge the difference in degree has become a difference in kind.
Individuals don't evolve. Populations gradually evolve over many generations.