The way Nxt does it is actually quite clever and solves the nothing at stake problem, there is more to it than this but the part of it that I understand is that choosing who forges next is every forger does SHA(last forger public key) Somehow how long ago an account forged and how many nxt owned by that account and information from the last block is factored in as well.  The winner is the one who has the largest hash.

So, in order to crack it, you need to use proof of work in order to find accounts that would have higher hashes for every block along the way.

This would take a network that does the same thing the Bitcoin network does but would be many magnitudes of order greater than the Bitcoin network to hash numbers in order to fast enough determine how an attacker would move around their coins into the right accounts in order to be able to author blocks.  And if it's not enough security, do SHA(SHA()) and you've squared the amount of processing power required by an attacker.  So an attacker would basically have to build a very expensive and long fork faster than the network does these light-weight operations.  He also needs to be able to plan 1440 blocks into the future in order to know where to place these Nxt in order for the network to accept his hash as valid.

There are some other things, such as not allowing block that have timestamps greater than X blocks ago.  And there is some other secret sauce that has been discussed behind the scenes that I'm not going to reveal here.

Point is though, this works in a completely different way to Bitcoin and the nothing at stake problem doesn't apply here because it works so differently.  Also, I believe there aren't 'forks' in the same sense that regularly pop-up as they do in the Bitcoin network because forgers aren't submitting their 'fork' to the network that they can re-write(short of that ultra high processing power), they are submitting their individual block.

Basically we aren't getting much further into the details though because they want to implement it and having working first before telling our competitors the right way to implement POS.  There are a number of people who have reviewed it however and it works.

You want to further discuss it, I recommend checking out this thread: https://nxtforum.org/general/how-does-nxt-fix-the-nothing-at-stake-problem/