The 24-year-old completed a move to Merseyside today, signing a four-and-a-half-year deal at Goodison Park.
"Obviously it's a different challenge for me," said Gibson. "I think being with Manchester United, you're winning every week, playing well every week.
"Obviously coming to Everton is a great club, it's a big club. They don't win as many games as Man United do but I'm looking forward to the challenge.
"I think it was time for me to move on from Man United and I think Everton's the right club for me to be at. With the Euros coming up I need to play games if I'm going to get in the Republic of Ireland team, so being here and playing every week will give me the opportunity of playing for Ireland as well."
The Toffees are believed to have paid up to £2m for the midfielder, which is the most that they've spent on a player in three years.