Ramsey waved goodbye to the Gunners fans following Sunday's 1-1 Premier League draw at home to Brighton.
Ramsey, sidelined by a hamstring injury, was unable to make a farewell appearance but was presented with a token of Arsenal's appreciation after the full-time whistle.
"Obviously very emotional. It's been a hell of a journey, 11 years of my life," a tearful Ramsey, who joined from Cardiff in 2008, told Sky Sports.
"So much has happened in that, when I look back. You can tell by my voice, how much it means to me. So I'm just grateful for the opportunity to play for this great club and spend so many years [here].
"I spent 11 years here, came at 17, a boy. And now I leave a man. I've got a family, wife and kids. So much has happened, I'm just grateful for the opportunity here.
"Obviously I'm really excited for the new challenge, new chapter in my life. But today it's all about this. This is where I grew up here, so I just want to take this in today, then let it settle a few days before I can really look forward and concentrate on that.
"I'm still only 28, so there's a few years left in the legs. So I'm looking forward to the future but today is all about this and sharing it with the fans."
For 11 years of service, for 369 appearances, for 64 goals, for 62 assists, for those Wembley winners, for coming back from that injury to achieve what you have with us, for EVERYTHING you’ve given to this club, we just want to say…— Arsenal FC (@Arsenal) May 5, 2019
THANK YOU, @aaronramsey! pic.twitter.com/EuU6gQ5U1q
Ramsey's focus will soon turn to joining the Serie A champions, linking up with Cristiano Ronaldo and a glittering array of talent in Massimiliano Allegri's squad.
"It's been strange. I'm still an Arsenal player and I've spent so much time here, most of my career has been here," added the Wales international. "So focusing on every game for Arsenal has been quite a challenge.
"I've managed to do it and give absolutely everything in every game, but now I can really let that sink in. I am going to be going to a club that's in the Champions League, playing with Ronaldo.
"I can let that sink in over the next few weeks and focus on that next challenge."
Petr Cech is also leaving Arsenal, with the veteran goalkeeper retiring at the end of the season, and he feels Ramsey will be missed.
"When you have someone who has spent more than 10 years in the same club, these are the core of the Arsenal dressing room, the Arsenal club," Cech told Sky Sports.
"Losing a player like that is always a big loss for the club, everybody. When he came from Cardiff, nobody really knew what a great player the club had brought in, and I think he gave everything he could all those 11 years.
"It's really a shame that today he could not be on the pitch, because everything he has done he deserved to be on the pitch today and maybe the score would have been better."