There are several variations of Pokémon eggs in Pokémon Go. Each of the eggs have a specific walking requirement you have to meet while inside of an incubator, and once they hatch, they give trainers a specific Pokémon inside. Not every Pokémon hatches from these eggs, and these are some of the only ways trainers can access certain Pokémon. Because there are so many different eggs in Pokémon Go, trainers now have access to bonus egg slots.
The bonus egg slots contain specific Pokémon eggs, notably the Red Team Rocket eggs and the weekly Adventure Sync eggs. These eggs will be held in these slots, allowing trainers to carry them there rather than have them take up a slot in the traditional egg inventory. This ensures the nine egg slots trainers traditionally have are only made up of the one, five, seven, and 10-kilometer eggs.
You can view your extra egg slots in the same menu as your other eggs. They will be at the far bottom of the menu. There are only three available so that players can hold up to 12 total eggs.
Given how much more space trainers need to play Pokémon Go, it makes sense that Niantic finds some middle ground to ensure player management doesn’t become too hectic. We could potentially see ways for trainers to add additional egg slots if more unique eggs come to the game. It varies on Niantic’s overall plans, and if more eggs come to the game, players will need more activities to acquire them.