In a very simplified way, proof of work is a way to roughly uniformily distribute the block generation "queue". In a network without identities how do you prevent someone creating hundreds of accounts to have a hundred times more chance to be next in line? You give them a hard problem, and now the odds of you being next in line is proportional to your computing power, which you can't multiply effortlessly.

Proof of Stake says that instead of distributing the work queue proportionally to the computing power you demonstrated to have, it does it proportionally to the amount of currency you have saved. It similarly prevents the attack where one could create infinite personalities to get in line, with different trade offs. In particular, beside the energy savings, it can be a much more scalable model, where you don't have to wait 10min in average for someone to solve the hard problem and instead you can know right away who are the next people eligible to generate the next blocks.