The asteroid hits but that isn't the end of the process. There is evidence to suggest it triggered massive volcanic activity and dramatic climate change. A lot of the extinction doesn't necessarily happen at the moment of impact, but in the many years to follow. It also helps to keep in mind that ecosystems tend to have a domino effect when they collapse: some species going extinct can lead to others going extinction.

Enormous large scale extinction follows the impact, dramatically changing the sorts of species we see in the fossil record. The whole thing is known among scientists as the KT event.

That being said, some dinosaurs did survive, and continued to evolve, and we see their descendants today in birds.

As for us, mammals prior to the KT event were relatively small (think rodent-like). But in the millions of years that follow there was a tremendous evolutionary opportunity with most dinosaurs gone. Mammals gradually evolved into new roles and diversified. Eventually one of those branches gives rise to primates. Out of the primate branch comes apes, and out of the ape branch, humans.

As for why our ancestors didn't go extinct, it's hard to say. But there is no reason to assume they wouldn't. The KT event is a massive extinction event but lots of things survived it.