Basically on a motherboard, there is a DC-DC converter located next to the CPU socket to convert 12V (that comes from the 4/8pin I assu,e) to the required voltage to the CPU. Lets just call this the CPU power stage. People refer this to all sorts of things from PWM circuitry to CPU phases etc ignore these as they make no sense whatsoever.
This "phase" normally refers to duplicating the main power components which are
1) The inductor/choke - square things that are located near those solid state capacitors next to the CPU socket.
2) Mosfets - flat square chips that you see near the square things.
So say a phase is made up of one capacitor/one inductor/2 mosfets. Now if you have 4 phases, youll see the above x4 all neatly located near each other.
They do this so that the "load" may be spread out instead of having to use components that are large and bulky to handle some of that load. Due to cost, heat, size and given electrical requirements, its very difficult to get away with a "single phase" converter which are pretty much the norm.
So you end up with motherboards with the CPU power stage containing 3~4 phases using cheap/average components to satisfy the CPU power requirements. Now obviously some of the high end boards use very good components i.e. get less hot/potentially last longer, but say having 12+ phase may not really be any different to say having 5~6. It would be actually be worse at lower loads in terms of efficency from experience.
The reason I say that is because that CPU power stage with lots of phases and good quality components could probably handle up to 300W of power! But if we look at what the typical CPU consumes in terms of power, alot of that is gone to waste. The designers probably know that they could have gotten away with 5~6 phases which would result in similar efficiency/temperature, but its probably all about the numbers/tick boxes/marketing slides.
Most boards will run for a long time if left in stock. When overclocked, components start to get stressed so typically higher end boards which have better components (which tend to have more "phases") would probably do well here compared to some el cheapo motherboard.