You still aren't getting it. There is a continuum between modern species and their ancestors. Even if the animals in between the modern species and its ancestor have all died out by now, at some point there was an animal that you probably couldn't easily put in either the old or the new species. The population at that time could be thought of as being in a transition. The transitional also breeds only transitional animals like itself.
If there is a transition, there is no need for humans to breed anything but humans and no need for apes to breed anything but apes. Instead, you have a human-like ape breeding other human-like apes. After some more mutations build up, the human-like ape becomes more human, like with a pelvis better adapted for upright walking, and a bigger brain. Even more time passes, and the brain is much larger than before, and upright posture is perfected. It isn't really clear where to draw the line and say "Okay, this fossil is a human," because the fossils are too similar. They definitely show a progression from more ape-like to more human-like, but you can't see where the ape ends and the human begins.
Paleontologists classify their finds into species partly just so that they have a handle to talk about them with. With fossils, it is impossible to tell what could breed with what. So if a fossil seems different from the known fossils in a significant way, it is grouped into a new species. There is often some debate as to where it should be classified, because it is hard to tell whether or not it is different enough to warrant a new species designation. So maybe Homo erectus could breed with Homo heidelbergensis, and the two are actually in the same species. There is no way for us to find out anymore.