The traditional way to mash your grains is in a container that has some way to cause the grain husks to form a filter. Coolers work well because they are insulated so your mash remains nearly the same temperature for the mash period. When that is over, the wort is drained slowly so the grain husks make this filter and the first bit of the wort is returned to the tun because it will have bits of grain and husks in it. That returning the wort until you get clear wort is called lautering. Once the tun is drained, you add water, stir up the grains and repeat the lautering and draining. That keeps the grain out of the boil pot.
Why is it done it a separate container? Because you need to remove the wort from the grains before you boil.
Another way of doing the mash is BIAB, where you line your brew pot with a fine mesh cloth and put the grains into that after the water is heated. It is preferred that you don't add heat once you add the grains because it is difficult to keep the grains and water mixed well enough that your temperature is the same throughout the pot. When the mash period is done with this method you pull the cloth bag with the grains inside out of the pot and let them drain.
