so where does Annie's power come from again?

Inborn magic akin to Lux's and Taliyah's, I would assume.

It may also have to do with the "fever" it said she had in her bio.

I think that it has something to do with her witch mother that gave her the bear.

I wouldn't say her story literally explains nothing about her powers:

Boram Darkwill’s last years on the throne were a time of great uncertainty for Noxus, and many with an aptitude for magic left the capital for the relative peace of more distant provinces. Gregori the Gray and his wife, a witch by the name of Amoline, preferred to demonstrate their Noxian strength by taming the borderlands, rather than partaking in the political intrigue of the noble houses.

her mom is a witch and "Gregori the Gray" sounds pretty wizard-y. I think it's safe to say her magic is genetic.