Checking for stale prices on a Chainlink feed is a great way to keep your dapp secure and future-proof. Sometimes, tokens change addresses, apps don't want to pay node operators anymore, etc, and it makes sense to have some fallbacks when such events occur.
The exact check you'd want to do depends on what you want your app to do.
The Chainlink Docs describe the two values as such:
updatedAt: Timestamp of when the round was updated
answeredInRound: The round ID in which the answer was computed
If your app is concerned about stale data (maybe more than 3 hours?), a check (in solidity) might look as such:
uint256 threeHoursSec = 10800;
if(updatedAt < block.timestamp - threeHoursSec) {
s_isPaused = true;
}
Where s_isPaused is some bool that controls certain functionality on your contract.
Checking answeredInRound is legacy from the fluxMonitor edition of Chainlink price feeds (which price feeds don't run off anymore) and it can be ignored.