It's unclear exactly how Google will ensure that only students sign up for this offer, but at a guess, they probably require you to use a school email address, or provide some other form of proof. Regardless, assuming you are indeed a student, you can claim your three-month trial by visiting the official promotion page and signing up.
As is always the case with free trials, YouTube Premium will begin auto-billing you once the trial is up, so if you don't have any interest in actually paying for the service, be sure to cancel it before then.
So, what does YouTube Premium actually offer? Is it even worth trying? While we can't definitively answer the second question for you, there are some neat perks for subscribers. For starters, you get access to YouTube Music, letting you listen to pretty much any song you can think of (though, as with other streaming services, song licenses do expire from time to time) with no limitations or ads.
Further, a Premium plan gives you access to an ad-free YouTube experience, as well as background play on mobile devices; meaning you can continue playing videos or music while your phone is locked, or while other apps are in focus.
The final perk worth noting is the ability to download YouTube videos. Of course, it's already possible to do this by using third-party tools and websites, but Premium makes the process legal, safe, and a bit more straightforward.
At any rate, once your trial ends, you'll be given special promotional student pricing if you do choose to stay subscribed -- you'll only be paying $6.99 per month instead of the $11.99 a month you'd normally need to shell out for Premium. However, the catch is, you'll need to provide "annual verification" of your student status to stay locked in at that lower price point.