In both literal and figurative sense: to give one's all; the ultimate in generosity

My grandmother used to tell me that during the war she would deny herself everything so that she could feed her children

La nonna mi raccontava sempre che durante la guerra si toglieva il pane di bocca per dar da mangiare ai suoi figli

He is a very generous person and is very into social problems. He would give the shirt off his back to help the poor

È una persona generosissima e attenta ai problemi sociali. Si toglierebbe il pane di bocca se solo potesse aiutare tanta povera gente

This hyperbole appeared in Tobias Smollett's novel of 1771, Humphry Clinker, and has been employed ever since