The Atletico striker spoke to Youtube channel Que Partidazo.
"His dream was to play with (Lionel) Messi and with (Luis) Suarez," said Costa.
"We all know the player that (Griezmann) is. As a player you always want to experiment and it's normal that he wants to play there and try to see if he was an important player as he was at Atletico. Griezmann wanted something new and that he has at Barca."
Costa added: "I didn't try to convince him to stay but if he had asked me I'd have told him that each person has to live through different things.
"I left Chelsea when I saw that it was the moment to go. To leave a team doesn't mean you're ungrateful to the club, just that the moment arrives to live through another experience."