Be quiet, you chatterbox.

Literal

Be-silent-[command-なさい], this chatterbox.

Two things going on. 黙りなさい is the ~なさい command form (a softer-than-imperative order, common from parents and teachers). The trailing このおしゃべり is a vocative — calling someone 'this chatterbox' as a direct address — using この + a noun the way English uses 'you idiot,' 'you fool.' このX is a recognisable insulting/admonishing vocative pattern.