How can I explain eczema to my friends and family so they know that I'm not contagious?

Genetics. Eczema is typically genetic and can have many triggers (food, environment, contact allergy etc) and there are many studies to suggest certain structural protein dysfunction resulting in eczema. Eczema itself is not contagious.
Inflammation of skin. Eczema is a skin eruption that is the result of inflammation. The inflammation causes scaling of the skin, much like a sunburn causes redness( which is inflammation) and then the skin peels ( much like scales or flakes from eczema). Eczema is in the class of skin conditions called dermatitis. They are reactions rather than infections. Tell your friends you cannot 'catch' someone's allergy.