I'm not sure if I missed something or not but how do you capture? ;-;

To capture a pet you need to buy one of these snares at the market (the market a Esior's Haven has them). Then when you are battling a pet you can access your snare from this satchel button. Click on the snare and then click on the sylesti that you want to capture. If the snare worked a message should pop up in the battle log saying that you successfully captured that pet and the pet will be in your stables. All traps except for Masterful Snare Traps have a chance of failing.

adding on:
Snares (outside the grove, but that's only relevant once you've finished the plot) only work in capturing pets of equal and lower rarity to themself. So a green (lesser) will only capture white and green starred pets, blue (standard) white, green, and blue, etc

And the three possible outcomes:
Success- you will be given a link to the pet in the battle log
Failure- the trap fails, the battle continues and you can try again
Failure- the trap fails, the pet you tried to catch flees the battle

Thank you both!