It means that if a problem or a disagreement or other difficult situation becomes so bad you have to take strong action and deal with it.

It is a farming idiom, meaning to reach a point where action is required. Cabbage, lettuce and broccoli, among other vegetables, cannot be harvested until the full head is formed. To achieve something is to bring plans and actions to a head. This is the literal meaning of the ancestor of our word achieve, which was borrowed from the French in Norman times. They made the verb achever out of the preposition a-, "to", and chief, "head": "to bring to a head."

This phrase also alludes to the medical sense of waiting for the head, the tip of an infection in or on the body like an abscess, a boil, pimple or other pustule, that is about to suppurate (break or burst). [Mid-1500s]