Can I get more info? How does the player gain this benefit? Is it a magic item? If so, what rarity does it have? Or is it an additional feature tied to the subclass?

Regardless, I don’t think the "high risk, high reward" mentality translates well into 5e, considering how different the system is compared to Vermintide or other video games.

A barbarian doesn’t have reliable ways to attract enemy attacks (aggro) or redirect damage toward themselves. They also lack built-in healing outside of potions and other magic items.
Even if we set that aside, being at low health isn’t truly a "risk" in 5e — In fact, it’s often the optimal way to play: there’s no difference between having 1 HP or 100 HP, and healing tends to be a waste of an action since enemies usually deal more damage than a player can recover. Rather than spending a turn healing, it’s almost always better to eliminate the source of damage instead.

That doesn’t mean your proposed feature is ineffective! It would significantly boost the player's damage output. However, it wouldn’t reinforce the "high risk, high reward" mentality you’re aiming to recreate from the original source.

I see two possible approaches to accomplish your goal:

The first is to provide a way to aggro enemies or redirect attacks.
This method gives the player a choice to take more damage, reinforcing the desired play style—increasing personal risk for the team's benefit.
Then, I’d introduce a weapon that improves damage rolls (not attack rolls) based on how much health the player has lost.

Here are some ideas:

Interpose:
When a creature within 5 feet of you is hit by an attack, you can use your reaction to interpose yourself, taking the damage instead of the target.

Provoking Roar:
When you enter a Rage or as a bonus action while raging, you can unleash a powerful roar. Hostile creatures of your choice within 30 feet must make a Charisma saving throw (DC = 8 + Strength or Constitution modifier + Proficiency Bonus).
On a failure, they have disadvantage on all attack rolls against creatures other than you until the end of your next turn.
You can use this feature a number of times equal to your proficiency bonus. You regain one use when you finish a Short Rest and all uses after a Long Rest.

Bloodied Fury:
Weapon (greataxe), uncommon (requires attunement by a Barbarian)
When rolling damage for this weapon, you can double your Rage damage bonus if you are Bloodied.

The second option would be to give the player a cursed weapon with powerful benefits and a corresponding drawback.
This approach allows the player to choose whether to wield a cursed item that enhances their damage but increases the risk factor. Unlike simply "taking more damage in combat by playing as intended", this creates a risk beyond normal gameplay mechanics.
An example:

Fiend's Maw
Weapon (Greataxe) +2 (rare, requires attunement by a Barbarian)
This weapon grants a +2 bonus to attack and damage rolls.
Cursed: Crafted by a vile sorcerer and tainted with demonic hunger, this weapon devours the wielder's restraint in combat.
While attuned to this item, you have disadvantage on Intelligence, Wisdom, and Charisma saving throws while raging. Your Rage damage bonus is doubled, or tripled if you are Bloodied.
If the curse is removed, the weapon's blade turns into stone, and the weapon becomes a nonmagical greataxe.

The last option would be to simply give the player an existing magic item of your choice. Again, I don't think any of them promote the "high risk, high reward" flavour that you are trying to accomplish, but they are effective.
An example would be either the Bloodrage Greataxe , a Berserker Axe, or the Groveltrash