How did they do that? Very simple: cull the ones that go broody most often, and incubate as many eggs as possible from the ones that go broody least often. Continue for at least a century.Most production breeds have had the broodiness bred out of them. How exactly did they do that. Breeders can't even manage to get the colours right in the majority of breeding programs let alone breed out something as fundemetal to a species survival as reproduction which is what ensures the survival of the species.
I think they have done a very good job at selectively breeding chickens that do not go broody in their first year of life under the conditions that are provided to commercial chickens.
Change the conditions (like to a backyard flock) and you get a bit more broodiness, but there still are major differences between how many Leghorns or Sexlinks go broody, and how many Silkies go broody, when they all begin as chicks from a hatchery and they are all raised the same way in the same flock.
(Since commercial chickens are only kept to a certain age, I also assume the selection has less effect on broodiness at older ages.)
Evidence of effectiveness: I have a book on poultry breeding & management from 1925.
The author appears to be mostly working with Leghorns. Every broody is marked with a legband, then put in a broody breaker cage. The author says not to breed from the hens that have 3 or 4 bands by the end of the season. Leghorns? Going broody that often?!
So either something else has drastically changed and that has affected broodiness (maybe housing or feeding?), or there HAS been a major shift in the genetics of domestic chickens, by selecting strongly against broodiness.