I am working on my foundations of computer science homework but see this one:

Initially, I try to draw a diagram like this:

(not the answer key)

, which is what we always do in class.

But it turns out the answer gives me something like this:

I've never seen this before, and still don't know how to solve my homework problem even after seeing the correct answer.

So could you guys tell me what this diagram is, how to draw one myself, and how to turn one NFA to DFA like the answer did?