The Following paragraphs are taken from Drake on Steam.
To get fertilized eggs, the first step in v219 is to get two tamed dinos of the same species together. Not all Dino species will support babies yet in v219, but most will. You can tell if your Dino species supports babies because it'll say on their HUD "Enable Wandering to Mate". So bring a lovely male and female (tamed) together, set them both to "Enable Wandering", and most likely pen them in to ensure they are kept quite close together for a time. When sufficiently close, beating hearts will appear over their heads, and the mating bar will appear on the female Dino's HUD. Both Dino's HUDs will also indicate who they're mating with, specifically.
When the Mating bar is complete, she'll drop a fertilized egg. She won't be able to mate again until a subsequent Mating Wait interval has passed, which is indicated on her HUD. (Males can mate as often as needed.)
The Fertilized Egg itself only incubates when dropped, not in Inventory. When dropped a Fertilized Egg won't spoil, but to incubate it must be kept at a certain temperature range depending on the species (this also factors in insulation). If outside of this temperature range, it won't make incubation progress, and it will lose "Fertilied Egg Health" over time and be destroyed if zero. This is all indicated on the HUD when looking at a fertilized Egg, and its description also tells you its Parents.
After being fully and successfully incubated, the baby dino will pop out of the egg. Babies initially have extremely low health and carrying capacity and food, so you need to carefully feed and take care of them in their first moments of life, or they'll soon die. They dynamically can carry more, gain more HP, and food capacity, as they mature over time. Babies are unclaimed initially, so you should "Imprint" on them to claim them after birth.