Se is reflexive, yes. Fame went to his own head. (Se lava los dientes: He washes his own teeth)
Le means "for/to him". Le traje un regalo a Camilo (I brought a present for him)
If it were talking about a group of people, say, a band, the sentence would have been: La fama se les subió a la cabeza. (Them)
Or only about myself: La fama se me subió a la cabeza (Me)
¿Me estás llamando? (Are you calling me?)
If you give a look at the search bar you may find other threads explaining it far better than me though.