Fig: to make someone arrogant or vain; to lose all sense of proportions

He's acting weird after the success of his first record. He thinks he's a musical genius and doesn't accept any critics. I'm afraid he let success go to his head

Dopo il successo del suo primo disco si comporta da stupido; si crede un genio musicale e non accetta critiche da nessuno. Ho paura che l'improvviso successo gli abbia dato alla testa

I like her a lot because her working in Hollywood didn't go to her head

Mi piace molto come ragazza perché nonostante lavori a Hollywood non si è montata per niente la testa

All those praises for his scientific research went to his head and now he thinks of himself as a new Einstein

Tutti quei complimenti per la sua ricerca scientifica gli hanno dato alla testa e adesso si crede un novello Einstein

Originally it referred to the effects of alcohol, which makes you dizzy or drunk, then it started to be used in a figurative sense to indicate other situations