To emote is to express emotion, particularly when you're acting in a film or a play. If an actor doesn't emote, the audience can't tell how he's supposed to be feeling in a scene.
When people emote, it sometimes appears overdone or melodramatic. Taking a feeling and exaggerating it so that an audience can understand the character's emotions is what actors do when they emote. The noun emotion came first, and the theatrical verb emote followed in the early twentieth century. The Old French root word is emouvoir, or "stir up," from the Latin emovere, "move out, remove, or agitate."