An incubator can only hold a certain number of eggs, and once that number is reached, it will no longer accept any more. The maximum number of eggs that an incubator can hold is based on its level.
Level 1 incubator can hatch 1 chick
Level 2 incubator can hatch 2 chicks
Level 3 incubator can hatch 3 chicks
Level 4 incubator can hatch 4 chicks
Level 5 incubator can hatch 5 chicks
There is one other factor that can affect the number of chicks you can hatch in your incubator: friendship with Chickens.
Each type of chicken has a friendship level, and the higher your friendship level with a particular type of chicken is, the more likely you are to hatch a chick of that type.
By increasing your friendship level with Chickens you're increasing a possibility of hatching more eggs.