What does the `的` mean in `我是九点去上班的`? In the reply to 你是什么时候去上班的?, what does the 的 in 我是九点去上班的 mean? I think that without the 的, the sentence makes sense in and of itself, right?

And the meaning is that I start to work at 9:00 am...., right?
These other answers are good, but they're missing the crucial element, which is why you have the 的 in that sentence. It's because 是, roughly translated into English as is, am, are, is a linking verb that is expecting a noun. If you said 我是九点去上班 that would essentially mean, I am go to work at nine. Which, of course, doesn't make sense.

However, 的 takes whatever is before it and turns it into an adjective, thus making "go to work at nine" an adjective. You're probably thinking, "Where's the noun, then?" Well, the noun is implied. That means that 我是九点去上班的 translates into I am a go-to-work-at-nine-type of (implied noun), or rather, I am a person who goes to work at nine.

Make sense? It sounds a little convoluted, but that's what's going on.