The selection uses three phrases, silent treatment, break character, curtain call. The one you ask about is a verb phrase to break character; the other two are noun phrases.
The teammates decided to give Ohtani the silent treatment as a "joke" (rather than immediately heaping praise upon him). Then they broke character (gave up this role of being silent) and gave him great praise. Then Trout told him to take a curtain call, which in this context means to go out onto the field and tip one's cap to the fans. As Edwin notes in a comment, the phrase break character is from the world of acting.
Perhaps the sports writer does too much ("goes overboard") in employing three such related phrases in such a short span of text.
Again, break character is a verb phrase; if it helps, you can write it as to break character. By contrast, the other two phrases are used with verbs, usually 'to give someone the silent treatment' (or, here to get the silent treatment) and 'to take a curtain call', just as in the story.