In proof of stake a single node decides what the next block added to chain is. This node gets the block reward and any transaction fees. They don't have to do any calculations, they simply get to decide, according to a set of rules, which transactions are in the next block. (Ethereum complicates this a bit by having multiple nodes "bet" which blocks will be accepted, instead of just one node choosing the block)

In order to decide who gets to be this node, people Stake coins. The probability that you are the chosen node is proportional to the number of coins you have staked. If you are chosen, and you do something against the rules, like double spend coins and try and add this to the next block, then you lose your staked coins.

So instead of rewards being proportional to hashing power, they are proportional to staked coins.

This is still being tested, but it's going pretty well.