You might be misunderstanding functors since they are basically tangential to classes. Every generic class `F<T>` where we can lift functions `T -> S` to `F<T> -> F<S>` is a functor.

Think of lists as example, they are generic over a type and we can apply functions over the base type by applying them to each element.

Functors are a basic building block of category theory. Category theory is basically the essence of composition so using functors leads to easy to reason about and extremely composable code. This also extends to things that build on functors like monads.