The 29-year-old Dutch striker played for the Gunners for eight years and won the FA Cup and the FA Community Shield.
"Arsenal is a great club and I always had a really good feeling, but after eight years sometimes you feel that you need a new challenge," The Guardian quotes him as saying. "I know why I went and I'm sure it's for good reasons.
"If you look at the last couple of months, it explains it all. I'm seeing things again that I didn't see for a long time.
"When I first came to Arsenal years ago [in 2004], I saw things from [Dennis] Bergkamp and Thierry Henry I'd never seen in my life before. Now I'm seeing those things again and that makes me really happy."
Van Persie added that he feels "like a kid in a sweet shop" at United.
"It's like a kid in a sweet shop," he said. "It's like, 'Yes, this is nice, this is good, this is having fun, this is enjoying yourself.'
"All these little things added together make a big difference. It makes me feel that I'm surrounded by champions."
Van Persie has scored 18 goals in 23 Premier League matches for United so far this season.