This saying is said to come from the American West, where the signal for a fight was often just the drop of a hat. It may have an Irish origin, based on something like "he's ready to fight at the drop of a hat" which in turn may be followed by "roll up your sleeves" or "take off your coat" ie items of clothing are involved in the start of fights.
If you do something at the drop of a hat, you do it suddenly and easily, often without any preparation.