What does Genesis 19:6 mean?

The prior verse demonstrated why Lot was so insistent that these two travelers stay in his home for the night instead of sleeping in the city square. Lot's offer of hospitality and an early departure was meant to protect these travelers from the depraved men of his city. Lot even suggested the men leave early, probably to keep their very presence a secret (Genesis 19:2). Unfortunately, the men of Sodom realize they have visitors, and they surround Lot's house, demanding that Lot release the men to the mob in order to be gang raped (Genesis 19:5).

Assuming Lot does not know yet that these men are angels in disguise, Lot demonstrates a considerable form of bravery in this verse. When the men of the city demand that he turn over the strangers so they can gang rape them, Lot actually goes out to them. Even more, he goes out and closes the door to his house behind him, cutting off his own escape from this violent mob. His pleas in the next verse are courageous, given that they're delivered to a warped crowd of would-be-rapists.