The main connection is, as others have stated earlier, physics and astronomy. The show could also have been named  String Theory, Quantum Theory, Relativity Theory etc. But The Big Bang Theory simply sounds better as a name for a tv show. In other words such title is bound to attract more viewers than e.g. The M-theory.

As has been pointed out, many of the show's characters are scientists, specifically physicists. The actual Big Bang Theory is an astronomy/physics theory, so it makes some sense to use this as a title for a show about physicists, though I agree that the connection is a bit tenuous. 

To me, the real connection between the show and its title is rather generic. It's a show about nerds who happen to be scientists and the Big Bang Theory is a recognizable scientific concept. 

I really don't think the show's creators were very concerned with relating the show to the actual scientific basis for the big bang theory. I think they wanted to create a humorous show around a group of exeptionally smart but socially inept characters. The difference between the character's IQ's and social abilities drive the comical situations in which they find themselves. The title itself is just a way to relate this idea to viewers, since almost everybody is at least a little familiar with the idea behind the big bang theory.

Besides the fact that the characters are scientists, the shows opening song gives you a clue to how it is related.  The "Big Bang" was the beginning of the universe, and thus the beginning of people.  The rest of the song describes human accomplishments, leading up to the present.  

One way the title relates the content of the show The Big Bang Theory to the actual Big Bang Theory is that many of the characters are theoretical physicists who study scientific theories relating to particles and space.  Leonard, Raj, Sheldon, and even Howard, all work on projects related to space at the university.  

The show's theme song lyrics "and it all started with a big bang, Bang!" also suggest that the comical events of each episode as well as the series often derive from a central beginning, like Leonard moving into Sheldon's apartment and becoming friends with him.