A basic premise about people is that they are all different. So, when answering this question, there will be many different answers. Let me give you a number of common reasons why students want to go off to college.
First, students want to go off to college, because they want an education and get ahead in life. Many people have been inculcated to believe that college will give them more opportunities. So, they go to college for their futures.
Second, some students go off to college for the sheer desire to learn. This is different from the first reason, because education may not be a means to a goal, but a goal in itself. Many of these people become teacher or scholars.
Third, some people go off to college for social reasons. They want to get away from home and experience independence. This is not a bad reason, because living alone can be a very important step toward adult life.
Fourth, some people go away simply to get away and be a twenty year old and experience life. This, too, is not necessarily bad. College, in my opinion, is a special time of life as you spend four years with peers.
Finally, I am sure that there are other answers, but the above four are some of the most common.