There's an idiomatic tradition in English of adding figurative phrases like "his eyes out", "her ears off", "their lungs out" and so on after intransitive action verbs.
They all have the structure [ possessive adjective + noun + particle ]. The only particles I can think of that are used in this structure are "out" and "off", but there may be more.
They all act as adverbials of manner, modifying the action verb.
They have the function of emphasizing the extent of the action.
So, to "bawl your eyes out" means to bawl hard and for a long time, as if long and hard enough to make your eyes come out -- but not really. To "talk someone's ears off" means to talk a lot, as a chatterbox does at a party. To "scream your lungs out" means to scream loud and for a long time, as sports fans do at a championship game.