Two-factor authentication, also known as 2FA, adds a secondary layer of security to your important online accounts by demanding a separate, unique code whenever relevant activities, like a password change, take place. Codes are typically sent via text message or email, although some games use standalone "authenticators," but regardless of the specific method, it ensures that even if someone gains access to your password, they won't be able to do much with it.
It's a hassle sometimes (I get tired of having to use my Steam authenticator whenever I need to log back into my account), but 2FA works, and numerous publishers and developers have been encouraging (or even requiring) that their users enable it. Electronic Arts is now nudging Origin users in that direction with an offer of a free month of Origin Access Basic, the subscription-based service that offers unlimited access to games for $5 per month, to everyone who turns it on before the end of October.
To activate 2FA (EA calls it "Login Verification"), log into your EA account and hit the "Security" menu button on the left. In the middle of that screen you'll see a "Login Verification" option, and what you want to do at that point (you've probably got this figured out already) is click the "Turn On" button. You can opt to have verification codes sent to an email address, texted to your mobile, or relayed via a voice call. There's also an authenticator app option which is supposedly the fastest way to confirm your identity, although I use the text message option and they roll in immediately when requested, so I'm not sure how much of an improvement you'll actually get out of it.
The free month will begin on or about November 1 (it might take a few days for the wheels to start turning) as long as you've got 2FA enabled on Origin before the end of October. If you're already an Origin Access Basic subscriber, you won't be billed for November, and if you're an Access Premier subscriber you'll get a free month of that instead. And if you've already got 2FA enabled? Free month for you too. The full details on EA's free month of Origin Access are here.