The game opens up alot more once you get an Aspiration, and as far as I know, Aspirations serves as Jellyfish Egg's means of creating some form of a character arc for the player character. Once you get one or more, the open narrative structure opens up alot more - and they may lead to different changes in your current run.
For example, if you defeat a goblin in self-defense, your character can gain an Aspiration in relation to hunting down goblins to the last and finding out where they come from. You get activities on hunting and tracking goblins, gaining equipment and skills in relation to them. Another example is gaining an Aspiration in relation to becoming a Gladiator which requires you to watch a single match related to Gladiators and have the money to watch said match for it, arenas will start having narrative options where your character can attempt to join a match.
Each Aspiration gives your character a means to gain certain skills. If you do a combat-related one, you gain combat skills and some other skills that can possibly lead to different Aspirations.
Understand that with no starting Aspiration, not many people are even going to find out that the game can open up. I would suggest having an option of starting out with one Aspiration or not starting with one based on the player character's starting attributes. Of course, make sure starting without an Aspiration is the recommended choice, as you want players to discover on their own. Players may become too reliant on having one from the very start and may never go out of their way to find out.
Players may never find an Aspiration that can change the narrative options, as some of them are too subtle for the differences to see.