Can you explain what a broodmare sire is please?

The correct way to describe a horse's sire (father) and dam (mother) is to say "the horse's name" is "by the sire's name" and "out of the dam's name", then you say the dam's sire "by the dam sire's name".  For example:
"Secretariat" is by "Bold Ruler" out of "Somethingroyal" by "Princequillo".  So Secretariat's father is Bold Ruler, his mother is Somethingroyal and his mother's father is Princequillo. 
The "dam sire" is also spoken of as a "broodmare sire" and can be an important consideration when buying or breeding a horse.  See http://www.bloodhorse.com/horse-racing/thoroughbred-breeding/sire-lists/broodmare which is a ranking of broodmare sires.