The Hatchery allows you to hatch any eggs you obtain that do not hatch immediately. Specifically, these are the eggs found by Toothless in Explorations. These need to be hatched in the Hatchery before they can be placed. You can find the Hatchery next to your Meade Hall.

The Hatchery starts with 1 egg slot by default, and can be expanded to 3 egg slots max. The time it takes for the egg to hatch depends on the dragon and its rarity. Once the egg hatches after the time shown, the dragon will emerge and it can be placed on your Berk.