This silver trophy "Armed to The Teeth" requires you to stay armed for an entire level.
This trophy is really easy and you can knock this one out immediately on your first level. The first couple of levels are more of an introduction/tutorial to the game.
In order to arm your character you have to hold the SQUARE button.
I would recommend you choose the very first level on "Gunpowder gardens". As soon as you selected the level, hold the square button, collect all the other animals and attack the alien.
That should net you the trophy; really easy!