Miners must find a hash below a target — achievable only by trillions of random guesses. The winning number is trivial to verify and enormously costly to find, anchoring the ledger to physics: rewriting old blocks means redoing their work faster than the honest network extends the chain.
PoW also solved fair issuance: coins go to whoever spends resources securing the network, not to insiders. Alternative designs (proof of stake) secure by wealth instead of work — a different set of tradeoffs and, bitcoiners argue, a different trust model entirely.