How do you fill out the AniLog?

The unit of progress is the species, not the individual. Your first catch of a kind you have never caught before is the one that puts a new entry in the book. Catching a second, a tenth or a fiftieth of something already registered does nothing for the log, so if filling it out is the goal, you are hunting for variety rather than volume.

That makes catching itself the whole of the system. Capture is what advances the collection, and anything that helps you capture helps you fill the log. Traps can be set into the terrain to assist, which matters most for the species that will not sit still long enough for a straightforward approach. There is no fee involved anywhere in this. Aniimo is free to play with no upfront purchase and no pre-order pack required to download it at launch, and the optional purchases sit alongside the collection rather than inside it.

The AniLog is what unlocks sealed promise eggs

The log is not just a checklist you tick for its own sake. Sealed promise eggs cannot be hatched straight away at launch. Each one first needs its Aniimo's form unlocked in the AniLog, and the way you unlock that form is by going out and catching the basic version of the creature yourself.

So an egg you are holding can be dead weight until the matching catch is on the books. If you are sitting on a sealed promise egg and wondering why nothing happens, the answer is out in the world rather than in your inventory. Find the basic version, catch it, and the egg becomes usable.

Not settled: Whether hatching a sealed promise egg adds its own AniLog entry, separate from the basic version you caught to unlock it, is not settled.