Fostering with animals means that the animal is up for adoption, but instead of staying at a shelter they spend their time living with a foster family in their home. The family is only taking care of them until they get adopted, so they won’t stay forever (unless it’s a “foster fail” where the foster family decides to adopt the animal themselves), but it helps the animal get used to living with people in a home. It’s useful for puppies and kittens to learn how to be good house pets at a young age, for elderly animals that need special care there isn’t time or staff for at the shelter, for traumatized animals that need a quieter and calmer place to recover from whatever difficulties they faced before being adoptable, etc.