Moving up in renown gives you a character point to increase your abilities, but it also gives you this game's version of charisma - your renown is a bonus you apply to social interactions. You leverage your fame and reputation when interacting with NPCs. You can also add your renown to one roll every turn; you are the legendary Gorom the Barbarian! You will not be defeated! GRR.
However, you also spend those hero points (like Karma in FASERIP) to add a die to a roll. You are trading growth for survival. As a solitaire game, Resolute needs several mechanisms to help the player character surivive. However, there is another benefit to this; you can lose hero points by having a battle go really bad, or by making bad decisions, or by doing sketchy stuff (and getting caught), or by just having a run of bad fortune. You get to keep your ability bonuses, but you don't get to keep the renown bonuses; I LOVE this, because now Strider is just Aragorn who has lost a ton of renown, and is fighting to reclaim it. Over in the corner there is Baldar; I believe he used to go by Baldar the Bold, but he nearly died in combat with the Wyrm of Elder Flame, and he has never been the same... he is but a commoner now, though I believe once he was a legendary hero of great renown. The game actually has a mechanism for this sort of stuff; it's actually central to the whole thing.
Your character sheet is largely dedicated just to a list of the deeds you have accomplished - you keep a record of all of your accomplishments to share over a round of ales (that you paid for in order to earn a hero point and increase your renown... you sneaky devil you).