How exactly does Barbarian rage work in 4E? Is there a prerequisite they have to meet to be able to rage, for example only after using certain powers, or can they rage whenever they want to?
I know of a Berserker Barbarian in the Essentials line, but I'm not familiar with it. I'm assuming you're asking about the PH2 class.
Unlike in the previous edition of the game, now raging is no more something you get by itself, it's part of some powers instead. Whenever you use one of the barbarian's daily attacks (they all feature the Rage keyword but Rage Strike), your barbarian character is considered raging until the end of the encounter.
This means you need to choose which battles are rageworthy, because you'll usually have less daily powers than encounters.
There are two different Barbarian classes with two different mechanics.
The Barbarian from Player's Handbook 2
The Barbarian Class from PHB2 rages by using its daily powers that have the Rage word in them. To quote the Compendium:
Thus to get the most utility out of a Barbarian's Rage you should use it at the start of what looks to be a big fight.
The Barbarian (Beserker) from Essentials
The essentials line also has its own version of the Barbarian, the Barbarian (Berserker). The Berserker class works as both a striker and a defender and the way the Berserker enters their rage reflects this. The Berserker starts out in their defender role (defender level damage, aura based punishes), and must rage to enter their striker mode (1d8 added to attacks, extra effects on martial at-wills):
The mechanics of the Berserker (and the Essentials line in general) allow you to be stronger on average, between fights rather than having 1 or 2 fights a day where you blow everyone out of the water.
My personal opinion: Both feel like barbarians, but from a mechanics perspective you have to be a much more thoughtful about when to use your rages as the regular barbarian vs. the Berserker subclass. Barbarians do more damage from a math/optimization stand point than Berserkers as well. I feel that the Berserker class feels more like the idea of a barbarian from a play standpoint though because you rage every fight and get to feel the benefits of raging every single fight. You don't question if you are going to rage, only when.