The 25-year-old joined the Midlands club for a fee that could rise to £6million on transfer deadline day.
Having joined the Toffees at the age of 10, Anichebe's rise into the senior side came under the tutelage of Moyes, who left Everton for Manchester United at the end of last season after 11 years in charge at Goodison Park.
"David Moyes has been such a help for me. He has been like a father figure," said Anichebe, who made over 100 appearances for the Toffees following his debut in 2006.
"He brought me through when I was just a boy and he has stuck with me through thick and thin. It was a wrench and there were moments even when I was driving down when I thought, 'is this the right decision?'"
Anichebe's final appearance for Everton was as a substitute appearance against Norwich on the opening day of the season.
"It was difficult but I thought it was the right time to leave," he added.
"Everton is changing and the club is going in a different direction. They have got a new manager in there now, with new philosophies.
"I was sad to see David Moyes go but that happens and you have to move on. I felt after a good latter stage of the season, maybe I should go somewhere else for a fresh start."