After reading various accounts afterwards, including Collins' interview earlier this summer in Cleveland, I've come to this conclusion (obviously could be very wrong).
Collins' performance was inconsistent. Not terrible, but bad enough that Mike Lombardi and others called him out. Was there freelancing? Probably not, just making more bad reads than anything.
After the 2nd Buffalo game, the Pats reached a breaking point and decided to make him part time until he redeveloped his consistency. Collins was not a locker room problem at all to that point, but I think the team made a projection as to what they thought would happen if they did that -- and they thought that Collins would start to ***** and moan for being benched during his big contract year, and that he had enough respect in the locker room that potential complaints could spill onto his teammates. The Pats were still only projecting - no actual issue had risen - but likely projecting based on past experience with other players (Adalius Thomas, Randy Moss) and their understanding of Collins' personality and locker room influence.
At that point they decided that it wasn't worth the risk. They felt it had to be all or nothing - either keep him as a starter, or if he wasn't going to be a full time starter, then he needed to be gone. So they quickly acted and got the best offer they could - which they thought was fair for someone they viewed as a part time player (and thus they certainly viewed as not being worth the required money for a contract extension). The fact that they could use the trade as "shock value" to the team on what happens when you are inconsistent was an added bonus.
All speculation, but it's my interpretation of what happened based on the reports.