To get enthusiastic or worked up; to become very into something or someone (colloquial). Can mean to raise excitement or to have a crush/lose one's head over someone.
コンサートでみんなで歌って会場の熱を上げた。
At the concert everyone sang together and raised the excitement in the venue.
彼女はあの俳優に熱を上げていて、話すと目が輝く。
She's crazy about that actor; she lights up whenever she talks about him.
試合で選手が熱を上げてしまい、ミスを連発した。
The player got carried away in the match and made a series of mistakes.