Why did Groundhog Day take place on Groundhog Day?

Apparently, it was mostly arbitrary. From a TCM article about the film:

Some viewers may be surprised to learn that there was no major reason why Groundhog Day was the holiday chosen for the film's backdrop. In an online interview with [the film's] writer [Danny] Rubin, he explains: "There were many reasons that Groundhog Day was a good arbitrary choice. It was a good choice because it's in the dead of winter. That made good sense for the story since the main character was stuck in his darkest day. It made sense that the character would come from out of town, and that the character was predicting the weather.... It's also an 'unexploited' movie holiday. The reason it became Groundhog Day was that I got the idea right around that time, and I happened to be one of the few people outside Pennsylvania that knew about it."

Rubin further elaborated in this Telegraph article that he just picked it out of a calendar:

When I sat down to begin writing this screenplay the first thing I knew I had to figure out was which day was repeating. To begin, I simply opened the calendar and the first holiday I came across was Groundhog Day, February 2. Yes, it was that simple.

In America people are vaguely aware of a holiday called “Groundhog Day”, and that somewhere there is a large rodent whose emergence from his burrow tells spectators whether or not spring is going to come early. But because of a writing job I had done years before for a Pennsylvania phone company I was one of the few people outside of that state who knew that the groundhog festival took place in a small town called Punxsutawney.

By choosing Punxsutawney I had a small, claustrophobic place for my character to get stuck. I also had a reason for my character, a weatherman, to go to the town: to cover the groundhog festival. I even had a name for my character: Phil, which I had borrowed from the famous groundhog.