Capturing animals is done by using the net launcher, a tool made from 2 scrap, 4 plastic, 1 metal ingot, and 2 bolts. You can craft net canisters for ammunition from 1 explosive powder (smelted explosive goo from poison-puffers), 4 stones, and 4 rope.
In order to capture an animal you have to be as close to the animal as possible as the net does not launch very far. You are able to capture llamas, goats, and chickens on large islands and there will be 1 - 4 of any combination of these animals per large island.
It should be noted that currently there is a bug in the game where if you have two people on one raft, each animal will count for two animals captured instead of one.