How did childbirth evolve?

Answer 1

Well the hominids that didn't produce children by childbirth died out ...

Childbirth seems to be something that early mammals got right and seemed to do well with.

The offshoot branches of mammals that laid eggs still exist. They are called monotremes (see the wiki page) and the current living examples of these are the platypus and the echidna. These are "prototherians".

Mammals giving birth to live young are called "Therians" and of these we are in a subgroup called "Eutherians", that is to say placental mammals.

There are many more therians than prototherians. Particularly the eutherians (placental mammals) seem to have got it right as there is an abundance of species and many species of eutherians have significant populations today.

Nevertheless, giving birth to live young is not exclusively the province of mammals. The wiki page on Viviparous birth says "Vivipary is best developed in placental mammals, but also occurs in many reptiles, some amphibians, crustaceans (e.g. Daphnia), scorpions, insects (aphids, the tsetse fly, some cockroaches) and a few fish."

Oviparity (the strategy of reproduction and birth by means of a shelled egg from which the young hatches) would seem to tbe the older strategy compared with viviparity.

Either oviparous mammals gave rise to viviparous mammals or viviparous pre-mammals gave rise to viviparous mammals (and then monotremes diverged early).

The former line, being simpler, may well be the answer.