How do you upgrade an Aniimo's core skill?

The upgrade is one-and-done. Each Aniimo gets a single core skill upgrade, ever, so a Prismatic Stone spent is a decision you do not get to revisit on that creature. Plan around which Aniimo you actually intend to keep in your team rather than whichever one happens to be in front of you when the stone drops.

What you get for it is a stronger version of the skill the creature already has. The upgrade raises its damage and adds utility on top, so it improves something already in your kit rather than teaching a new ability. Capability Fruit is also part of upgrading a core skill, alongside the stones.

Materials that do not touch the core skill

Star Essence and Star Sands are for stats and potential. They will not improve a core skill no matter how many you pour in, and the reverse holds too, so a Prismatic Stone is not a way to fix a creature whose numbers are lacking.

Capability Fruit does double duty. It is also what raises an Aniimo's rarity tier, taking an Elite up to Perfect and granting plus one Initial Potential in that creature's recommended stat. If you are holding fruit, you are choosing between two different upgrades for it.

Not settled: Because Capability Fruit feeds both the core skill upgrade and the rarity tier upgrade, how much of the core skill upgrade it accounts for against the Prismatic Stone is not settled.