The breech in the headgasket allows hot combustion gasses to be blown directly into the water jacket, adding more heat than the cooling system can dissipate, adding volume by adding gasses, and adversely affecting the ability of the cooling system to disspate heat. Much as a hot pan will cool far slower in air than in water, your engine's heat will dissipate slower if the coolant is aerated.
The headgasket seals the oil passages between the block and the head also, and if an oil passage is involved in the breech it can allow movement of oil into the coolant with the combustion gasses.
If you do not repair the headgasket, your engine's life is going to be very short. The breech will erode quickly and grow, the head will inevitably warp, and you rengine will be history.
The reason you need to refill the radiator directly is because the gasses being blown into the water jacket are displacing the coolant. When the engine heats up the air pushes cooolant to the reservoire, and when it cools after you shut it off, it retains the air content, thus not drawing all the coolant that it should back into the engine. Opening the radiator cap and fillling it there allows the air out and fluid in to replace it.