ANSWER: Before we explore what might have led to the incest between Lot and his two daughters, it is importance to remember that overall God considered him a righteous man (2Peter 2:4 - 9). It was, without a doubt, incredibly difficult for someone like Lot to be obedient to God in a city wholly given over to pursuing its lusts.
Most people are somewhat aware of what happened to Sodom and Gomorrah. Lot, his wife, and two daughters are saved when God sends two angels to make sure they are out of Sodom before the city is punished for their wicked ways. The request of Lot to have the family flee to the nearby small town of Zoar, instead of the mountains, is granted (Genesis 19:18 - 22). As they came into Zoar, God brought brimstone and fire upon the two sinful cities.
The stay of Lot in Zoar was likely not very long. Although he begged the two angels to not send him into the mountains but instead let him seek safety in Zoar, he soon leaves the city and heads for the mountains (Genesis 19:30)! Once he and his family are in the mountains they find a cave to live in. It is then that the older sister tells the younger one the following.
Our father is getting old, and there are no men in the whole world to marry us so that we can have children. Come on, let's get our father drunk, so that we can sleep with him and have children by him (Genesis 19:31 - 32).
The daughters proceed to get their father Lot drunk with wine then have sex with him. They do such a good job at getting him inebriated that he does not know what they are doing to him (Genesis 19:33, 35). The two women become pregnant and eventually give birth to boys named Moab (father of the Moabites tribe) and Benammi (father of the Ammonites).
Before the incestuous events involving Lot, the family experienced some awesome and heartbreaking events in a very short period. They had to run from Sodom in order to save themselves and then saw the aftermath of Sodom and Gomorrah destroyed. Then, not only is the wife of Lot turned into salt, they must flee yet again and end up living in a cave. Would any of us not feel a bit distressed, confused, and afraid of what might happen next if we saw and experienced what they did?
It certainly seems the young women panicked when they were at the cave with their father. This may have led to their confused thinking, which led them to make conclusions that were not accurate, which were then used to justify their impulsive reaction to the situation.
The Bible states that the daughters justified having sex with Lot because they believed all men were dead (Genesis 19:31)! This, however, shows they acted impulsively. They forgot they left the city of Zoar, which surely had some men in it even after God had completed punishing Sodom and the surrounding cities (Genesis 19:29 - 30).
On the surface, the desire of the daughters of Lot to continue the human race after Sodom and Gomorrah is certainly noble. Their behavior, however, was based far more on their skewed thinking brought on by fear and panic. As for Lot, he sinned willingly, but not willfully. He succumbed to human nature and put himself in a compromising situation.