Why do soap molecules not break down your skin cells when you wash your hands?
The outer most layer of the mammalian epidermis (cornified layer or stratum corneum) is composed of 15-20 layers of dead cells called corneocytes, which are basically dead keratinocytes filled with keratin intermediate filament cross-linked other proteins as well as some lipids. As keratinocytes differentiate into corneocytes (a process called cornification), the plasma membrane is replaced by what is called a conified envelope which is composed of cross-linked structural proteins and some lipids (more details in here). 

Surfactants in soap do interact with components of the cornfield layer such as keratin but not in the same way they would with the lipid bilayer of a typical plasma membrane. Outcome of such interaction depends on the kind of soap as well as amount and duration of application among other factors, but even a normal soap, upon normal use on a normal skin, may cause some keratin denaturation. But the cornified layer is actually continually shed (process called desquamation) and replaced by keratinocytes that proliferate in the inner most layer of the epidermis.  

You can find more details on effect of detergents on skin here and here. You can also find useful info on skin structure here and here.