Many of the things that nannies teach kids are basic, such as the alphabet, colors or counting. These are skills that need to be constantly reinforced at first in order to be learned. Since the nanny is with the childen most of the time, she is able to teach them these basic skills through practice and various activities. If practiced daily with the nanny, these concepts are much easier to grasp.
A major social skill for smaller children is learning how to share. Often, when children are very young they have never had to share unless they have a sibling around the same age. It is important that they learn this skill prior to going to school so they are able to socialize in a way that is friendly and fun for the rest of their classmates and their teachers.
Another social skill that children must learn young is manners. "Please" and "thank you" are terms that must constantly be reinforced. The nanny teaches your children these skills since she is doing things for them all day. Through practice, they learn when it is appropriate to say please, thank you, sorry, and other very important terms. They should be reminded of these things constantly when they ask for something so they can ask properly and say thank you.
Nannies also teach children many things about the way the world works. Children are naturally curious and will ask a lot of questions when they don't understand something. It is the nanny's job to be able to answer their questions in terms which they will be able to understand, and then build upon that knowledge with more specific information as the child continues to learn and grow. These things seem like common sense to nannies, but it takes a special person to be able to explain them in a way that makes sense to a child.