Amy, Cream, Rouge & Blaze (along with several of Sonic's other friends) are time frozen by the Time Eater.
During Sonic's birthday party, a mysterious entity called the Time Eater appears. It knocks Sonic unconsious & then proceeds to suck his friends into time vortexes. This results in them having their life force (and color) drained from them, and their bodies time frozen as a result. Sonic, now stuck in the White World, has to team up with his past self, Classic Sonic, in order to fix the damage that the Time Eater has wrought and save his friends.
Each of Sonic's friends is positioned to the lower-right of each level gate. They remain colorless & time-frozen until both Acts of the Zone they are outside of are completed; at which point a cutscene will play that will show the character being restored to normal, followed by a thank you that will differ depending on which act in that zone you completed last.
Noting each character location: Tails is in Green Hill, Amy Rose is in Chemical Plant, Knuckles is in Sky Sanctuary, Cream & Cheese are in Speed Highway, Rouge is in City Escape, Espio is in Seaside Hill, Blaze is in Crisis City, Vector is in Rooftop Run & Charmy is in Planet Wisp.