Putting your own money on the line
Proof of Stake takes a different approach to the same problem. Instead of competing on computational work, participants called validators lock up, or stake, their own capital as collateral for the right to propose and confirm new blocks. Validators are generally chosen to propose blocks through some combination of how much they've staked and other selection rules, rather than through a computational race.
The security argument shifts from physical cost to economic cost. If a validator acts dishonestly, for example by trying to confirm two conflicting versions of history, the protocol can detect that behavior and destroy some or all of the validator's staked capital, a penalty generally called slashing. The validator is putting their own money where their vote is, and standing to lose it if they cheat.