The task here is simple you need to build and bait a trap and then successfully capture an animal with it.
The easiest way to complete this is to follow Blane's quest. He will send you to a nearby cave to capture rats. Before you go he'll want to craft some traps. You'll need a wooden shingle and a paper clip to build one. He'll end up giving you five for the quest.
After the traps are crafted grab them from the stock pile and head out to the cave. The destination will have bait items available in the cave, check the cave for loot before setting down your traps. After you've grab the misc. food around the cave go to your inventory screen, click on a trap and chose place. Put the trap along the side of the room and put any of the food you wish as bait.
You can see the rats running around the screen, if you are stomping around they will run away. So turn off your light and crouch (left bumper.) After a rat runs into your trap they will either be captured or possibly trigger your trap and escape. If they escape just place another trap. If they are captured don't forget to loot them, just walk up to the body.