A couple of reasons:

1. They don't trust people to read everything. A lot of readers drop off before the end of an article just because their attention flits away. Pull quotes are a way of saying, "Here's something coming up that I think is interesting. If you are interested, you should keep reading."

2. A lot of people have trouble with long runs of samey text. Some see it as boring, others as imposing, others as hard to navigate, but for whatever reason, long runs of text are simply hard to read for a lot of people. So pull quotes are a way to break up the text without resorting to vaguely relevant cat pictures.