In addition to keeping the heat inside the boiler as the burner cycles on and off during normal operation, they'll also stop winter air from coming down the stack and freezing the water in the boiler tubes (potentially bursting them) whenever the boiler is not in operation. But don't just take our word for it …
“In their effort to find practical means of reducing fuel consumption and slow down the rise in space heating costs, both the industry and the government have recognized that certain devices added to heating systems could yield energy savings. One such device is the electrically operated automatic vent damper (stack damper). Its purpose is to close the appliance (boiler, furnace, water heater, pressure washer) exhaust vent every time its burner shuts off, in order to limit the amount of warm vapors (air) escaping through the stack (vent system) or chimney.” - Source ASHRAE Case Study