Here’s my understanding of the system so far:
They get to kill someone! Basically in vitro fertilization became more and more popular and sex came to be more and more for pleasure than for reproduction, to the point where most people choose to be sterilized when they go through puberty. Rapidly declining birth rates forced the government to implement an incentive to bear children, and that incentive was the “殺人出産” system, where anybody (and I’m guessing anybody male or female here due to artificial wombs), can choose to bear 10 children and as a reward get to kill a person of their choice. (It also seems that the sterilization process is surgically reversible).
I believe it’s seen as a noble sacrifice because of the amount of danger and effort associated with giving birth ten times, and the societal need for a larger population. This is where Ikuko seems uncomfortable with the shift in attitude toward murder from the past, because under the 殺人出産 system it’s glorified rather than resented, and the school system seems to indoctrinate young students with this altered morality. I don’t think the manner in which the birth-giver gets to kill their victim has been elaborated yet–I don’t know whether all they have to do is name a person and that person is killed by the government, or if they are just granted legal immunity for committing a murder on their own, or something else entirely.
Additionally, the penalty for committing murder outside the system is (depending on your point of view) even more severe than the death penalty. It’s now a birth penalty, where criminals (even male via artificial wombs) are forced to repeatedly give birth.
Also, birth-givers do not seem to have to raise their children themselves. Instead the they entrust the children to “centers” where they are presumably cared for and can be adopted.
Please let me know if my understanding is off or if I’m missing any key details. I definitely can’t presume I’ve understood it correctly myself.