This is a curious practice implemented with the quarterings for my grandfather. For
each of the earldoms of Warwick and Gloucester, an early earl bears both their own
arms and that of the earldom's.
In the case of Gloucester it is clearly the arms of Clare with the arms of William FitzRobert, second earl of Gloucester and who had three heiress daughters one of whom married a Clare who later became earl possibly de jure uxor. But this can also be seen as a normal case of adding a quartering with the marriage to an heiress wife although this is not how the College treats this.
But The Warwick pair is not as simple as this. The College labels them as 'Newburgh' and 'Warwick'. The first of these in The Dictionary of British Arms, Medieval Ordinary, volumes 2 pp. 273-4 where it is variously named as Newburg and Warwick. The second is difficult to find and the two nearest blazons are on Volume 2, page 201:
These two blazons do not make it any clearer where the second arms came from, nor what rule was being followed.
My conclusion is that this is not a rule of quartering that has a firm foundation. However both these pairs of arms are, or appear to be, arms that are quartered by the usual rules of inheritance so they are included in the quarterings of the Kingmaker.