The heat pump is only moving the heat. Burning coal doesn't move heat, it makes heat by destroying coal. Because the heat pump isn't making the heat it is able to move the same amount of heat from somewhere else (say, the outside air) into your home, using less energy than you'd need to make that heat inside your home.
Because you can't destroy heat (thermodynamics) for refrigerators we have no choice, we must use heat pumps. But once you get good at this technology, it's just better regardless of whether you want to move the heat away (Air conditioning, refrigerator) or towards you (heating a home).
This is how heat pumps claim seriously > 100% efficiency in many cases, they're comparing the efficiency of moving heat against making it.
It's important for people to know that at least when it comes to heating, "100% efficiency" is actually the worst possible efficiency. That's just thermodynamics. You can only get better than that.
No, you can get much worse as well when considering heating in general.
Heating efficiency is a measure of useful heat, so while of course energy is conserved in the universe, it is not 100% efficiency as measured, where we consider the consumed energy versus the successful heating of the home. Heat lost to the outside is inefficient.
With natural gas heating, efficiency used to be around 50% as the exhaust gases were quite hot. They just had passive chimney flues driven by natural convection. Improvements to recover more of that heat brought efficiency up to 80% and now approaching 95% in the most modern configurations that I've read about. They now require mechanical ventilation to remove the exhaust and also have condensate drains because the heat exchangers cool the gases so much.
Of course, something like a traditional wood fireplace is much worse than even those old gas furnaces, with a vast amount of energy going out the chimney.
With electrical heating, if you consider transmission losses you can see that the effective home heating is less than 100% of the generated power. The power lines and equipment warm the outside too.
Another benefit of the high COP heat pumps: in reducing the total power consumption at the home, it also brings a proportional reduction in those losses outside the home...
While this is true in some sort of physical sense, it's not really useful for home heating. The problem is that while burning stuff makes heat, that heat probably isn't where you wanted it, and you need to capture the heat and move it to where it's useful. Unfortunately burning stuff produces exhaust gases, and those too are hot, that is they contain some of the valuable heat, but the gases themselves are dangerous and unwanted.
Modern boilers achieve about 94% efficiency. That is, if we make 1MJ of heat energy by setting stuff on fire, the boiler transfers 940 kJ of heat energy into hot water. The remaining 60 kJ is warm exhaust. You've probably seen the exhaust from people's homes if you live somewhere with such boilers.
The heat pump is only moving the heat. Burning coal doesn't move heat, it makes heat by destroying coal. Because the heat pump isn't making the heat it is able to move the same amount of heat from somewhere else (say, the outside air) into your home, using less energy than you'd need to make that heat inside your home.
Because you can't destroy heat (thermodynamics) for refrigerators we have no choice, we must use heat pumps. But once you get good at this technology, it's just better regardless of whether you want to move the heat away (Air conditioning, refrigerator) or towards you (heating a home).
This is how heat pumps claim seriously > 100% efficiency in many cases, they're comparing the efficiency of moving heat against making it.