I am pretty sure it was because he was not happy with the contract he was offered. I am guessing since Miami drafted him, they had control over him being able to go to another team. So ol' Joe decided to skip town to Canada.
Found this in an article on the net:
Although he was drafted out of Notre Dame to play for the Miami Dolphins, a contract dispute prompted him instead to pick the Toronto Argonauts of the Canadian Football League. He played there in 1971, '72, and '73. Miami, meanwhile, went on to the Super Bowl three straight years, losing in '72 and winning in '73 and '74, leaving Theismann and millions of other people wondering why he had taken such a hard line with the Miami organization.
I think we drafted him in the fourth round. I'll bet he felt real good after spurning us and then we went to 3 straight Super Bowls!! What an idiot! Of course he probably feels vindicated, because he did throw 2 TD's and beat us in a Super Bowl!