Ping
Instead, He kept Peter and the other disciples from sharing this news widely because He knew the general Jewish populace was expecting a political form of salvation alone (v. 20). Perhaps broadcasting His messianic identity would have cut His ministry short, before its appointed time of completion, for the Romans were quick to squelch any potential Messiah and an uprising that people following the Messiah might instigate. Perhaps He kept His office as Messiah hidden from all but a few people so that He could accomplish the most important messianic work—atonement.
Or as human nature would have it, a sure-fire way to get the word out about anything is to say, "Mum's the word", or "Can you keep a secret?" or, "Promise not to say anything!"
Classic.