How do fire sprinkler systems put out fires?

Fire sprinklers have two major components that operate in response to a fire: the fire sprinkler heads which are connected via a series of pipes to a fire pumps. Water pressure is kept in the head by a series of valves in the pump. and is kept in the heads by little vials of colored liquid.

During a fire, the liquid inside the sprinkler heads heats up and starts to expand, eventually causing the vial to burst. Water that was kept in the pipes pours out of the sprinkler heads.

The water pouring out of the sprinkler heads causes the pressure in the fire pump to drop. In response, the pump kicks on and pumps more water from the municipal water supply to keep the sprinkler flowing.

NOTE: this is how "wet pipe" fire sprinklers, the most common types of fire sprinklers, work. Other sprinklers (dry pipe, deluge and pre-action) are slightly different.