The 28-year-old had been tipped to exit the San Siro this summer after a move to Juventus collapsed in the previous transfer window.
"The people who needed to do something to keep me here did so – and by that I mean everyone, from the club to the fans," The Express quotes him as saying. "Now it's up to me to prove my worth. They helped me and now I have to help myself.
"What's in the past is in the past. We have a new group of players now and a coach who is really hungry and determined – he shows us players that constantly.
"I'm here and ready to show put my skills to use on the pitch, just as he does it off the pitch."
Guarin has won 51 caps for Colombia and played for his country at this summer's World Cup.