He ended last season on loan to Lorient, where he scored one goal in 15 appearances and made only six starts.
However, he was named the most valuable player at the Toulon youth tournament last month.
That prompted United manager Sir Alex Ferguson to move in and increase his attacking options following the departure of Cristiano Ronaldo to Real Madrid.
Obertan told French radio station RMC: "I knew they had been following me for a few years but I did not think it was going to happen this season.
"We had no longer been in too much contact until the England Under-21s game where I played well for France and scored a goal.
"That helped things along. We spoke again, and I ended up signing.
"He [Ferguson] has put me at ease.
"He said to me that he knows my pros and cons. It's my potential that interests him. He has confidence in me. It is flattering and surprising.
"But I know I still have a lot of work to do to get to that [Ronaldo's] level.
Lorient coach Christian Gourcuff is not convinced Obertan will be a success at United.
"He certainly has some talent, but also some gaps (in his game)," Gourcuff, whose son Yoann is the playmaker for France and Bordeaux, told Europe 1.
"I hope he will succeed, but he will have to make the step up mentally."
Obertan will sit out United's tour of Asia, because he is suffering from a minor injury.
The club head off next week and play their opening match on July 18 against a Malaysia XI.
Three other matches have been scheduled in Indonesia, Korea and China.
Rio Ferdinand believes it will be ideal preparation for the coming campaign.
"The important thing about pre-season is getting some good training done, " the England defender told MUTV.
"One of the main reasons we have been successful in the last few years is because we've had a good base from pre-season.
"It takes a bit of time to get used to the humid conditions in Asia. But it is a good exercise to play against teams that pose different threats."