Final answer:
At level 13, barbarians gain the ability Brutal Critical (1 die), which adds additional damage on a critical hit with a melee attack. Other abilities mentioned are attained at different levels.
Step-by-step explanation:
At level 13, barbarians in Dungeons and Dragons 5th Edition gain the ability called Brutal Critical (1 die), as seen in the Player's Handbook. This ability allows them to roll an extra weapon damage die when determining the extra damage for a critical hit with a melee attack. This significantly increases the potential damage they can deal when they do land a critical hit.
However, the other abilities mentioned, such as Fast Movement, Feral Instinct, Extra Attack, Persistent Rage, Indomitable Might, and Relentless Rage, are acquired at different levels of the barbarian class progression.