Gallagher was pushed out of the club last month after being frozen out by new manager Enzo Maresca.
“Last season I had a good season with Chelsea, and you could see by (Mauricio) Pochettino playing me every game and showing his trust in me," he said. "The fans appreciated what I did on the pitch.
“I’m very grateful for all of that and very thankful to the fans, and very thankful to Pochettino for giving me that platform to go and express myself and show the best version of myself at Chelsea.
“Honestly, with Chelsea I have only got good things to say. Amazing football club, my boyhood club where I grew up through the academy. I leave Chelsea with really special memories.
“In football, things happen. As everyone knows, there was a lot going on over the few weeks that it was all happening with Chelsea and Atletico trying to agree a deal.
“Other situations came into it, but I was always confident the deal would go through because of how the manager (Diego) Simeone and the club spoke about me and to me. I’m really happy with the outcome.”
He added: “Chelsea make their own decisions and they’ve brought in some top players on long contracts. That wasn’t the situation with me but, in the end, what happened is good for everyone, so everyone is happy.”