Answer: Before we explore what might have led to the incest with his two daughters, it is importance to remember that overall God considered Lot a righteous man (2Peter 2:4 - 9). It was, without a doubt, incredibly difficult for someone like him 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 his 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.
And when the morning dawned, then the angels hurried Lot, saying, "Rise up! Take your wife and your two daughters who are here, lest you be consumed in the punishment of the city" (Genesis 19:15, HBFV).
The angels, after leading Lot and his family outside the city limits of Sodom, tell him again to flee.
"Escape for your life! Do not look behind you, nor stay anywhere in the plain. Escape to the mountain lest you be consumed" (verse 17).
Lot, fearful something bad will happen to him in the mountains, requests from the angels that his family be allowed to flee instead to the nearby small town of Zoar. His impulsive request is granted. God, shortly after the family enters Zoar, destroys the sinful cities of the plain with fire and brimstone (Genesis 19:24). During the destruction Lot's wife looks back at the sinful cities and is turned into a pillar of salt.
Fear Leads to Sin
The stay of the family in the city of Zoar was likely not very long. He soon leaves the city and heads for the mountains (Genesis 19:30)! He then finds a cave to live in. It is then that the older sister tells the younger one the following.
"Our father is old, and there is no man on the earth to come in to us, as is the way of all the earth. Come, let us make our father drink wine, and we will lie with him, so that we may preserve the seed of our father." (Genesis 19:31 - 32).
The daughters then 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 Lot's wife 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.
Impulsive Justification
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 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 allowed himself to be put in a compromising situation.