Fig.: to laugh secretly; to lough quietly to oneself
(informal)

The old patron of the billiard hall, feeling at ease, challenged the newcomer to a game of pool. He, a billiard champion who happened to be traveling by, laughing up his sleeve accepted the challenge

Il vecchio habitué della sala da biliardo, sentendosi a proprio agio, sfidò il nuovo arrivato a una partita a pool. Costui, un campione di biliardo di passaggio, ridendo sotto i baffi accettò la sfida

The Formula One leader is stuck in the pits due to a major technical issue, and his direct opponent is chuckling to himself

Il leader di Formula Uno è fermo ai box per un serio inconveniente meccanico e il suo diretto avversario se la sta ridendo

When Amalia, the school's most popular girl, invited shy William to the movies, all their friends were laughing up their sleeves, knowing that it was an April fool's joke

Quando Amalia, la più bella ragazza della scuola, invitò al cinema il timido Guglielmo, tutti gli amici se la ridevano sotto i baffi, sapendo che era un pesce d'aprile

The original expression “laugh in one's sleeve” was already in use in the 1500s. 
In the Middle Ages, people would usually wear big, loose sleeves and use them to cover their mouth to conceal their laughter