The confusion in this story is caused by the NRSV's overly literal translation methodology and its under adherence to that methodology in verse 31.
So there are two problems with the translation.
The first problem is that the Hebrew idiom דרך כל הארץ, which the NRSV translates as "the manner of all the world", is a Hebrew euphemism for sexual relations.1 Because the word "world" in this verse is used in a euphemism that stands on its own, independent of the verse, it has no relation to, and does not contrast with, modify or limit the prior "earth". In fact, what Lot's daughters intend to say is that there are no men left in the entire world to have relations with them so that they will bear children.
The second problem with the NRSV translation is that it uses two different words, "earth" and "world" to translate the same Hebrew word, ארץ ("aretz"), that is used in both instances. This variation in translation sets the stage for the misunderstanding.
In understanding this verse it is important to consider the narrative. Nothing in the preceding narrative indicates that Lot and his daughters are anything but righteous. There is an assumed stricture on incest, but up to this point in the text no explicit prohibition. There is an explicit command in Genesis 1:28 and in Genesis 9:7 to be fruitful and multiply. So in the narrative of this verse we can only assume that Lot's daughters would not have done what they did unless they were convinced that there was no other choice in order to fulfill the commandment, i.e. that there were no other men anywhere who could father children.
The NIV attempts to address the translation problem in this verse by both rendering the meaning of the idiom and translating the idiom literally:
One day the older daughter said to the younger, "Our father is old, and there is no man around here to give us children--as is the custom all over the earth.
The NLT chooses to render the idiom and forfeits the literal translation:
One day the older daughter said to her sister, "There are no men left anywhere in this entire area, so we can't get married like everyone else. And our father will soon be too old to have children.
The ISV both defeats the euphemistic intent of the idiom and renders the idom literally (!)
One day the firstborn told the younger one, "Our father is old, and there's no man in the land to have sex with us, as everybody else throughout all the earth does.
1. The Masoretic text on which the NRSV is based uses euphemisms and "clean language" in all except one instance when referring to sexual acts.