Fig.: to be alert and watchful in order to avoid being taken off guard and, thus, be ready to act for the best. To look very carefully for something or someone

The vacation package you've been offered is really cheap, but it sounds too good to be true. If I were you, I would be on the lookout for hidden costs

Il pacchetto vacanza che ti hanno offerto è veramente a buon mercato, ma mi sembra troppo bello per essere vero. Fossi in te starei in campana per eventuali costi nascosti

This is the decisive game and the home tennis player must keep his wits about him, as any mistake could be his last

Il tennista di casa deve stare in campana perché è il gioco decisivo e ogni errore potrebbe essergli fatale

 It's quite a rough neighborhood, and if I were you, I wouldn't go there. But if you must, keep your eyes peeled!

È un quartiere poco raccomandabile e se fossi in te non ci andrei, ma se proprio devi andarci, sta' in campana!

This is the decisive game and the home tennis player must keep his wits about him, as any mistake could be his last

Il tennista di casa deve stare in campana perché è il gioco decisivo e ogni errore potrebbe essergli fatale

Watch out, because if I find out that you have been cheating me, I'll make you pay dearly for it

Sta' in bandiera perché se vengo a sapere che mi hai imbrogliato te la faccio pagare cara!

One theory sustains that this idiom comes from England and it is in reference to Robert Peel who started the British Police Force. The nicknames for the police were "Bobbies" or "Peelers", peelers referring to keeping a watch out for lawbreakers.
Another theory refers peeled to "peeled fruit", so by peeling back your eyelids you can see everything clearly and not miss anything