The 29-year-old was signed from Fenerbahce for £6.8million in August 2011, but made just 26 appearances.
Santos feels the short-term move will work well for all parties.
"I am confident this is the best for both Arsenal and myself," he said on the club's official website.
"I just want to say a big thanks to all the people that helped me at this wonderful club.
"I wish all the best for the team for the remainder of this season and of course, I'll be following the team from Brazil.
Speaking in a separate interview with the London Evening Standard, he added: "The idea is to go there, play well and be able to come back strong. Do I have a future at the club? For sure.
"This is a loan deal and after this season we will see where everything stands.
"I always wanted to stay at Arsenal until the end of my contract. This is a club where I feel comfortable and honoured to play.
"However, when the club acquired another left-back it was clear to me that I will not play. I need to play, get my confidence back and Gremio is a good platform for this.
"I talked (with Wenger) and we decided together this would be a good chance for me to get on top of my game.
"He gave me a hug and told me Arsenal would always welcome me back with open arms.
"He did (want me to stay), but he wanted to see me happy and if I am not performing all the time, then I am not happy."
He added: "Things did not work out as well as I expected. I am not satisfied.
"Hopefully, when I am back, I will be on top of my game and then I will be able to help the team the way I want to."