"Going off to" somewhere means to leave and go to that place. It emphasizes separation, and is often used to describe a long-term trip or permanent move.
When used without an article, "university" is a generic term. It can be used to refer to a university that someone attends, or to the state of being enrolled in a university, or to a stage of life. Americans use the word "college" in the same way. (For Europeans, a college is more like a trade school, so don't use that word with them.)
"Going off to university" means leaving home to attend a university. For most students, this will be the first time they have lived outside of their parents' house. It's a big turning point in life.
Here are some American English examples of "college" without an article. (I'm not Canadian or British, so I won't try any Euro-style examples.)
I haven't seen a party like that since college. (stage of life)
I'm going back to college in a week. We should hang out before then. (place)
I want to go to college for engineering. (state of enrollment)