A failure in the ignition switch or the igniter can cause a stove to continuously click. Clicking can also indicate that the burner is dirty or that the igniter has been in contact with moisture.
Use a metal pin or straightened paperclip to clear debris from the electrode and burner holes. Do not use a wooden toothpick, which could break off during cleaning. If there is any moisture on the burners, wait for it to evaporate or it can be dried with a hair dryer. If cleaning and drying the unit doesn't stop the clicking, the ignition switch may be broken and need replacement.