You can buy and sell animals from this shop, you can also buy a few items for looking after them such as medicine.
When you purchase a chicken the Livestock Dealer will leave it in the pen behind his shop for you to pick up and take to your farm. With the cow he will deliver it to the farm himself.
When you want to sell your animals you need to lead/place them into the corralled area left of the shipping bin. Afterwards visit the Livestock Dealer and tell him you wish to sell the animal.