This is the best (and also approachable well-written) book on the topic that I've found:
https://bitcoinbook.cs.princeton.edu/My (possibly incorrect) understanding is that POW is computationally expensive because that large investment of computation is what creates a chain of successive blocks (the blockchain). This prevents someone from rewriting history of transactions on the public chain (which would allow them to 'double-spend' or to take their money back).
POW currencies are guaranteed to prevent this kind of abuse unless any individual entity is able to get more than 51%. There's an incentive in addition to this because corrupting the integrity of the network would also devalue the currency. Larger networks (like BTC) are harder to do a hostile take over of because it's harder to get that much compute (though mining centralization is a risk).
POS relies on some variant individuals 'staking' coins to enable transactions, this means putting them up in escrow sort of in the network (they are paid small fees for this based on how much they stake) and if abuse is attempted, the system takes those staked coins away. There are no mathematical guarantees outside of this incentive.
POS is not as standardized across different currencies so I may be missing important bits in my understanding.