The verb brillare in Italian means to shine or to sparkle, but in Italian slang, you can also use the adjective brillo to say that someone is tipsy or slightly drunk.

Personally, I think this can be explained by that sparkle or shine one shows in his eyes when ‘slightly drunk.’  However, the Italian etymology gods say something else. If I understood correctly, brillare is the ‘shining’ effect that wine has when it is generously poured into a glass and some spills out.

During my brillo research, I also discovered the word alticcio, meaning tipsy, or being merry [from drink]. The etymology of alticcio is also interesting, saying that it comes from a guy named Altetto who was apparently “somewhat altered for the excessive drinking of wine…”

Examples:
‘Sono brillo, non posso guidare.’ – I’m tipsy, I can’t drive.
‘Non ho mai visto Giovanni così alticcio.’ – I’ve never seen Giovanni that tipsy before.

Disclaimer: The Italian etymology website that I use seems to be [badly] scanned pages of an actual book, and furthermore, an etymological book that is pretty old in its style, so if I get it wrong or miss something, please excuse me!