The Netherlands star, 28, has joined the Turkish club on a three-and-a-half year contract after cutting his ties with Inter following a contractual dispute.
Inter asked Sneijder to take a cut in wages earlier in the season and, when he refused, made him available for transfer.
Now the former Real Madrid and Ajax midfielder says he wants to emulate the great Gheorghe Hagi and become a firm favourite at Gala.
"I realised how big the Galatasaray community is after arriving at the airport," Sneijder said. "I am really very happy and proud that I will serve such a big club.
"The derby match against Besiktas [on Sunday] is crucial. I recovered from injury a long time ago and continued training by myself. I am in good shape and if my coach picks me I want to play in the derby.
"It's a real honour for me to be compared to a legendary player like Hagi. I know how important his name is at Galatasaray. He was a legend and I came here to be a legend."
Reports said teams including Premier League Liverpool were also interested in the 28-year-old, who inspired Inter Milan to a Champions League, Serie A and Italian Cup treble in 2010 before helping the Netherlands finish World Cup runners-up.
"That was the best season so far in my life. The period after was a hard period," he said.
Sneijder struggled to maintain that success and was caught up in a contract dispute at Inter this season after failing to renegotiate his deal as the club sought to cut costs and being banished from the team.
The signing of Sneijder is a major coup for the Turkish league, which is more used to taking top players towards the end of their careers like Fenerbahce's Dutch forward Dirk Kuyt.
Turkish league leaders Galatasaray, who unlike Europa League contenders Inter are in the Champions League last 16, confirmed a deal was done after Sneijder announced the move on Sunday following days of pondering their offer.