Maybe it was a combination of getting away from something and being drawn toward something.
Even snowy, icy climates have their "summers" which are periods of a little warmer temperatures when there is some partial melting and some plant growth. That would provide food and water for some animals that humans might have hunted. Following animals into snowy regions during those "summers" could have introduced humans to a gradual adaptation to colder climates as they learned how to keep warm and find shelter.
I don't know of any human ancestors older than Neanderthal that lived in cold, snowy environments. H. Sapiens and our Neanderthal ancestors/cousins needed to have already developed the use of fire and a level of tool technology that made survival possible in such cold climates.
But why would they bother to adapt to such weather and CHOOSE to live there? Getting away from competitors for resources in their own species or even from another skilled hunting animal? Avoiding confrontations? How many wars are fought in Arctic climates when all your energy needs to be focused on daily survival and mutual dependence? A preference for fighting against natyre instead of each other?
Or maybe resources in their original habitat became scarcer due to climate change making some regions dryer. There is always water available in snowy, icy climates. Just melt some ice or snow.
I also wonder why some humans adapted to desert life. They just didn't want to move when their habitat got hotter and dryer?