Since Moses and his family were the only people left, they did what any normal human would do in this situation. Spread the population!
The technology of their previous society was mostly washed away since Moses was busy building a boat and his relatives were probably busy helping build it. Therefore, he and his family had to start all over again.
So Moses' sons Shem, Ham, and Japheth had their own children, who also had their own. This eventually turned into an almost infinite cycle of procreation until a dude said, "we should settle down somewhere," and another guy said "Good idea!". The different families split into different nations, with each their own language, and settled by rivers.
Why did they settle by rivers? Because every civilization is started by a water source.
A bunch of stuff happens, like the stone age, agriculture, all those things. After a while, these people infested the Earth, building their own empires.
Some of my favorite and most famous are:
Babylonian Empire: The big bad strong
Ancient China - Boom, crackle, gunpowder!
Ancient Egypt - Pyramids and stuff
Ancient Greece - They were pioneers in math and failing grades
The Roman Empire - They were big
The British Empire - They were bigger
ImagiNATION - Best nation EVER
These various civilizations are the foundations on which we stand today, and through their accomplishments, we have built a glorious and magnificent world. We truly are standing on the shoulders of giants.
TL;DR?
Civilization happened. There were some pretty big empires.