It can drive parents crazy when their young children constantly beg to do things for themselves. "Me do! Me do!" is their constant cry. While giving in to temper tantrums over such an issue is not recommended, allowing children to do these things and more for themselves is. Giving children a reasonable amount of responsibility builds confidence, competency and their sense of self-worth. In the long run, at least, it makes a parent's job easier as well.
Maria Montessori explains this concept beautifully: "To teach a child to feed himself, to wash and dress himself, is a much more tedious and difficult work, calling for infinitely greater patience than feeding, washing and dressing the child one's self. The former is the work of an educator, the latter is the easy and inferior work of a servant. Not only is it easier for the mother, but it is very dangerous for the child, since it closes the way and puts obstacles in the path of the life which is developing."
Further Information:
This concept of not taking initiative away from the child is related to the Church's "Principle of Subsidiarity". You can read more about that here.