He didnt sign with the Pats, he was traded to them by the Eagles. The problems was that he was strictly being used as a situational pass rusher with the Pats, and he didnt like it and mouthed off at the coaches.
Belicheck doesn't tolerate stuff like that, you dont question hoodie and how he runs the team.
So, he was suspended for "conduct detrimental to the organization" and told to pack his bags, he was done in NE. They would have flat out released him, but they were scared he would sign with an AFC contender like the Chiefs so they offered him up for basically free to any NFC teams and the Cowboys said yes.