Answer:
The term monogamous refers to a species that has only one mate at a time, as opposed to polygamous species which typically have more than one mate.
In the example given here, many males are being killed for their prized plumage. If this continues to happen, the species may die out. This is because once the female mates with a male, she will stay with him, mating for life. Many females will be left without a mate because the ratio of males to females is so low. And so, the number of chicks hatching will continue to drop.
This wouldn't be as big of a problem if the species were polygamous, because the males may have multiple mates at once (or in quick succession.) More eggs become fertilized this way, and the species may have had a chance for survival.