Landowning families had seals, fairly unique identifiers of each individual. Many of the early medieval landowners could not read, so it was convenient to use a seal as their mark on formal documents. The designs on these seals rapidly developed into coats of arms, to be used on the field of a tournament, perhaps to identify the competitors.
When the father died without sons, his daughters inherited his lands, usually divided between them. But male supremacy being what it was, any husbands took possession. Though unmarried daughters, or widows of course, could and did own land; such daughters had to have seals.
It was not always obvious in medieval times who owned what land, due to the feudal practice of holding land from a superior. So I speculate here that it was then convenient to the husbands of these heiresses to somehow use their father-in-law's seal to show that they had good title to the father-in-law's land. Two practices developed to show the involvement with the heiress' family:
What happens if a succession of heiresses are married, a new one by each generation? Well the arms of each may be added in. The first heiress to be married by the most remote direct male ancestor goes in quarter two, then the next most remote heiress of her male ancestors in quarter 3, etc. And more arms are added in some symmetrical way, thus losing the meaning of "quarter"; any spare boxes at the end get the paternal arms inserted. Taken to an extreme, here's an example.
In all this remember that it does not matter how many daughters there are. The children of each daughter may quarter the arms of their mother's father. All inherit their maternal grandfather's arms equally.