Stakers take act like miners, but instead of staking computation time through proof-of-work, they stake an amount of cryptocurrency which gets burned if they ever double-vote (and maybe if they vote for the losing chain, etc, depending on the scheme).

It's not very obvious how exactly a proof-of-stake scheme should work though. Many past attempts have been pretty flawed. I'm excited to watch Ethereum's progress in this area.

Not that I'm totally sold on it, or grok it fully, but from my rough understanding the very highest level (for Ethereum's Casper)...

The idea is that a double-spend attempt by a cartel of validators could be included in a new block as cryptographic proof to used take away their stake entirely ("slashing" it). Where a BTC miner would merely lose the cost of an attempted double spend block, and could keep mining more bad blocks; removing their stake completely is like burning down their rig.

In order to prevent that punishment, they'd have to control >2/3 of the staked coins on the network. Which means as long as total amount staked grows in value proportional to the network, this will be incredibly expensive. By adding rewards for staking, this further incentives long-term holders to stake part of their holdings, to secure the network.