To "keep a straight face" means to avoid laughing. You try to "keep a straight face" when something is funny but you're not supposed to laugh. This happens in situations like:

• when you're in a serious meeting but someone makes a mistake.
• when you're trying to play a trick on someone.
• when you're scolding a child who's been bad but they say something funny.

We mostly talk about not keeping a straight face or trying to keep one:

When he was talking, I tried to keep a straight face but I just couldn't.