Jose likes to keep one or two experienced heads in the team. More than what they can do on the pitch, it's also the influence they have on the dressing room. It's important. Carrick is a veteran who can impart a lot of useful intangibles (such as a "winning mentality") to the younger players. He is one of the only players remaining from our 2008 CL winning side. It's a wise decision to extend his contract.
Having older players around to provide guidance is important. It's why Fergie kept Giggs and re-signed Scholes. It's why Mourinho signed Zlatan and why he re-signed Drogba and leaned heavily on Terry, Zanetti and Ramos at previous clubs. Good decision to keep Carrick on board. In fact Carrick should be made club captain next season.
I agree (huh, never saw that coming). Regarding the captaincy, he might as well be the "official" captain, but given the very limited amount of game time he will, and should get (if he continues), he really won't be the on-the-pitch captain so to speak.