To capture a monster you need three things: first, you must wound the monster until it's near death (it'll have a skull icon near its marker on the minimap if you've properly researched it), second you must use a device to pacify the monster further (ideally a Tranq Bomb - it usually takes two to subdue the monster) and three, you'll need an actual trap (a Shock Trap or Pitfall Trap both work fine).
The run-down is pretty simple. For preparations, make some Tranq Bombs (Sleep Herb + Parashroom) and a trap (Shock Traps are the easiest - Thunderbug + Trap Tool). Once done, accept any assignment/quest/investigation to hunt any monster (save Elder Dragons, which cannot be captures) and fight like normal, being sure to investigate enough tracks ahead of time to max your Scoutfly Level, so you can track the monster's status. When it has a skull above its head, it's easiest to let it retreat to its lair and sleep, at which point simply run up to it, smash two Tranq Bombs into its head, then plant a trap under it. If done correctly, it should wake up just long enough to get trapped. This shortens your struggle considerably and grants you extra rewards, so as long as you go through the trouble to create the tools necessary, it's actually easier and more lucrative to capture many monsters than to slay them.
It's best to start capturing monsters as early as possible (ideally after you hunt the Barroth, where you should have the concept introduced to you), as you need to capture fifty monsters to earn this trophy/achievement. That's a long grind in itself, but if you start early enough... well the core gameplay loops involves killing lots and lots of monsters, repeatedly, so ideally you'll be earning this while you work towards other goals, like gathering materials or completing assignments and optional quests as you advance through the main story.