There are two ways to read this question. First, it might be asking us why the antiwar movement arose in the first place. Second, it might be asking us why it arose on college campuses in particular. Some of the reasons overlap, but some are different.
The Vietnam War in general started to become unpopular when large numbers of American troops started to be sent to Vietnam and the war did not end quickly. To most Americans, it seemed like the war should have been won quickly. When it was not, people started to question the way the war was being fought and even whether it was the right war to be fighting in the first place.
The protests against the Vietnam War popped up on college campuses first for at least two major reasons. First, college professors tend to be much more well-informed about foreign affairs and are typically more liberal than the average American. Because of this, college professors were more likely to be outspokenly against the war. The same was true of many students. Second, the students at colleges were of the age most likely to be drafted into the war. Therefore, the war had much more of an immediate meaning to them than to most other Americans.
For these reasons, the Vietnam War became unpopular and the protests against it started on college campuses.