Why would a professor (say in pure math) want to take PhD students?
They get to train the next generation of mathematicians. They get to establish a relationship with young students, guide new researchers when they are up-and-coming and those people become their students. For the rest of time those researchers will be the person trained by Professor Stella and that is intrinsically valuable for several reasons:


Some people like teaching. I love teaching, and if I leave industry and go into university research it will be largely because I want to teach. The ability to work with and mould young minds really appeals to me, and I get value out of knowing that I've helped people learn.
You develop an attachment, and a sense of pride, in your students. Their accomplishments become, not quite yours, but close to feeling like yours. If you’ve ever served as a TA, you’ve probably worked for a long period of time with a struggling student. Maybe you felt a twinge of pride when they finally got it, or got a good grade on an exam. When you’re an advisor, you get a lot more than a twinge.
They are part of your legacy. They will tell stories about being your student for another 20+ years after you pass away, they will toast you at banquets in their honor and thank you for making them the researcher that they are today. 


What a professor expects of you is that you try your very best to be a good scientist, that you do interesting work, and that you continue to carry on the torch of mathematics for another generation after they are gone.