How do you hatch sealed promise eggs at launch?

So the egg is not a shortcut past the catching. Whatever you sealed, you have to go out and catch the ordinary version of that same species first. That registers the form in your Anilog, and only then will the egg open. Until that happens it just sits in your inventory, which means a rare or awkward-to-find species can leave a promise egg parked for a while after launch.

What comes out has a Potential grade fixed on the same four-tier scale as a caught Aniimo: Common, Good, Elite, Perfect. Hatching is not a guaranteed upgrade over the copy you just caught to unlock the form.

A Sparkling Cube is the tool worth knowing about here. It guarantees a capture, so it can settle the basic-version step outright, and it also applies during egg incubation, where since the launch update it grants guaranteed Perfect Potential.

What testers were allowed to seal

The Bond Pact event let a tester seal one Aniimo, singular, in an egg to reclaim at release. Prismana Forms were specifically excluded from it, and Prismana and other special Aniimo cannot be sealed inside promise eggs as carryover rewards at all. Any Prismana Form you built up during the closed beta tests does not come with you to launch.

Other eggs do not work this way

The Anilog requirement is specific to sealed promise eggs. A Mysterious Umbrabow Egg hatches like an ordinary egg and produces the Umbrabow form, the darker shadow-touched variant of the species, with no catalog unlock standing in front of it. Sparkling eggs likewise hatch normally and guarantee the sparkling trait on the Aniimo inside. Prismana eggs can be hatched too, and can yield a perfect Aniimo.