Why did Lot's daughters sleep with him?

Answer 1

Genesis ch.19 "Our father is old, and there is no man in the land..."
After God destroyed Sodom, Lot retreated to a cave. His daughters sought to become pregnant.

They chose their own father for this purpose, either because:

they thought the whole world had been destroyed, as in the Flood (Rashi commentary);

or they meant that there is no man in this area and they didn't want to uproot the family and move (Ovadia Seforno commentary);

or they meant that there is no man of our status (ibid); since, as descendants of Terah, they saw themselves as nobility.

Answer 2

To get the answer read Genesis 19:30-38. To preserve Lot's lineage.