The expression “laughing up your sleeve” means, to laugh quietly or secretly. When you do this, you are ridiculing the person.
*Nagesh pretended to admire Sachu’s paintings, but we knew that he was laughing up his sleeve at her. *John looked very serious, but he was laughing up his sleeve at the terrible performance.
What is the connection between laughing and someone’s sleeve? I understand that in the 16th Century, it was fashionable for men to wear a shirt that had fluffy sleeves. The sleeves were so big that whenever an individual wanted to laugh secretly, he could hide his face in them and do so. This ensured that those around him didn’t know what the he was doing.