The trophy description is not very clear at all. Basically you will come to a point on the "Captured" level
. Because of this you cannot reload any weapon. You simply need to empty the clip of your gun into an enemy and you will discard your current gun then just pick up the gun the enemy has dropped. Rinse and repeat 20 times and the trophy should pop.
Effectively you need to pick up and use 20 different weapons during this part of the game.
The weapons don't have to be unique and you'll find that you are picking up the same few weapons over and over again.
It seems to still count it if you just change weapons without emptying the ammo clip but I emptied my clip every time to make sure I didn't miss out.
There should be more than enough weapons to pick up to achieve this as long as you try and change your weapon at every opportunity.