застенчивость usually implies being shy, awkward in terms of communicating to other people, social inhibition, and another similar word is also стеснительность.
if you seriously think about some Russian words you will often find the 'hidden' etimology (in fact not so hidden):
hence, застенчивость - as if a person is за стенами, behind the walls
стеснительность from стеснять, тесно, теснота - so the implication is 'tightness, tight and awkward social situation', the person being tight inside, socially inhibited