Berserkers were a type of Norse Viking that went into battle less armored than the others, wearing little more than animal skins rather than the metal mailcoats normal Vikings wore. They weaponized their anger to fight with more brutal ferocity, sometimes going so far as to even attack their allies.
This happened long before science was a thing, so we can only hypothesize and theorize how they did this. They wore the animal skins (usually bear and wolf skins) to gain favor from the gods, most notably Odin, and they likely consumed drugs or alcohol to put themselves into this state before a fight (certain genetics probably played a large role as to who could berserk as well). The lack of protection probably helped activate their adrenaline, the body's natural Limit Break chemical, as they sustained injuries during a fight, alongside being exposed to the elements.
https://www.youtube.com/watch?v=BdaZZanDI34
https://www.youtube.com/watch?time_continue=17&v=di_yrogI0io
https://www.youtube.com/watch?v=p-XbxdrTZEs
Your body is a lot more capable than you realize, but the brain naturally inhibits your latent DBZ-like powers because of the amount of strain they cause. During particularly stressful situations, you'd probably notice how you suddenly gain superhuman strength or you stop feeling pain so much. This is because your brain realizes that you might be in mortal peril, so it starts temporarily shutting down certain functions (usually everything related to digestion goes first), taking the long term risk of damage to save itself in the short term. Of course, this only lasts for a short while, lasting only a few seconds to maybe a minute or two at most.
https://www.youtube.com/watch?v=k8Jlq_E2eCg
Anyways, berserking was a class that was brought to tabletop games, namely D&D and Pathfinder (which is basically D&D). Berserking is a fighting style unique to the Barbarian, but as of Pathfinder at least, it's not exclusive to them. It basically means you get a huge buff in a small burst, increasing your damage, damage resistance, and sometimes speed with a few other bonuses here and there, with the consequence being you are exhausted for a while after.
Berserking in League has been around since the very beginning. There are at least six different champions who have a different aspect of berserking; if one champion had them all, they'd likely be the most broken thing in the game.
Darius: His is based on his own attack momentum. Getting five stacks of Hemorrhage on a champion supercharges his damage and maxes stacks on all future targets for a short duration. His method of berserking is probably the smallest example (and most stable), considering he doesn't exactly exhibit rage or rage-like tendencies like the other rage-based champions. It's more of a martial trance or being "in the zone" than a rage; Darius' method of fighting is using brutal precision, precision being the operative word there.
Renekton: Renekton is a more fantastical/video gamey definition of a berserker. His rage does little more than boost the potential of his skills, and it's less like a rage and more of a bloodlust, acting as the missing link between Darius and the rest of the berserkers.
Gnar: Probably the second smallest example, Gnar's rage powers his Hulk-like transformation, which has no real life counterpart, hence why it's further down on this list. He transforms by quite literally getting angry, turning into a gigantic rage monster and smashing everything in sight. It is blatantly a rage power, but again, it has no counterpart in real life.
Sion: The angriest guy in League, I think, Sion is the epitome of almost literal blind rage. He fights in lore like an actual berserker, attacking enemies and allies alike and sustaining a huge number of injuries during his (un)lifetime. His attacks and his weapon are completely unrefined, focusing more on brute strength than any actual technique; hell, most of his attack animations don't even use his axe, and only one skill actually uses the axe. His rage power is simply refusing to die even after he's been killed; this also powers his Soul Furnace passive, giving him more health and thus making it harder to kill him.
Tryndamere: A closer example of a berserker in League, Tryndamere's rage powers influences the effectiveness of his powers, increasing his crit chance and his ability to "heal" and ult himself (in real world explanations, he's basically using adrenaline to ignore his own injuries).
Olaf: The most authentic berserker in League, Olaf is the closest example to what a berserker was like in real life; lightly armored, beardly, and forgoing defense for extra bloodletting offense. Olaf's rage powers are more subtle than the others, making him fight harder the closer he is to death, increasing his attack speed and sustaining himself through bloodlust. His ult is the quintessential aspect of the infamous berserk charge; ignoring the return fire (though ultimately not defying death like Tryndamere or Sion) as he closes the gap with the enemy and hacking them to death before he dies from his own injuries; it's the offensive counterpart to Tryndamere's defensive (the not dying part) berserk.
Berserking is not just about getting really angry, but being able to control it in such a manner that you benefit from it rather than it outright crippling you (otherwise anyone tilting would suddenly be the most dangerous players in all of League rather than being the least threatening). It is possible to berserk in real life, but most don't know how. I am still learning how to berserk, but I have done it many times in the past, even during a League game. It makes you more aware, more focused, and you can "see" even where you can't, but it is a very hard note to land, and as I said, I'm still learning it. It helps that I have OCD and my brain is constantly active with one thought or another, so I can trigger myself more easily than most, but I am also in control of myself enough that I will never tilt into the bad end and become a liability to the team.
is doing is an over exaggeration of berserking, where the berserker gets stronger by channeling their anger into strength, making them go beyond their normal capabilities, thus resulting in bonus crit chance in game. By doing this, they seem to be invincible and look as though they can shrug off any wound, much like Trynds Ult and healing abilities.
He gets really angry as the battle continues
hence his passive "Berserker's Rage"
"They're ruining my good mood."
"Oh-ho-ho, they look angry."
When ulting Tryndamere:
"C'mere, Princess."
