Capturing is a game mechanic that can first be unlocked after obtaining Spider Silk in Area 3-1.
Once you have at least 10k gold and a SpiderSilk, you can craft the Trap, unlocked for 10k Alchemy Points in the 100 mL or 1 L tab of Alchemy.
To attempt to capture an enemy, right click on the enemy after crafting a Trap. This will use 1 Trap for every attempt and will tell you "Captured" or "Miss". You will start at a 10% base chance to capture normal enemies, and a 0% chance for bosses, until damaging to 5% HP or lower.
You can keep track of how many of each type of enemy you have captured in the Bestiary menu in the Explore tab. Each captured enemy increases the loot chance of items in that enemy's drop table with diminishing returns. The first 5 each provide a 1% increase, the next 5 provide a .25% increase, and so on. The bonus is multiplicative, so 102 monsters give a 27.13% total bonus and if something had 0.04% chance to drop it becomes 0.04%*1.2713=0.05%. Each captured enemy also decreases the time to loot that monster in the Bestiary, which can be automated with the Auto-Collect Bestiary Loot from the QoL 1 tab of the Epic Store for 5500 Epic Coins.