Animals have to be fed to be able to give you products. You can produce food for animals in the barn:
The barn, like any building, comes with 2 production slots. You can buy 2 additional slots for the total of 4. Choose the recipe that you want to produce by sliding up and down between various recipes. Once you choose the recipe, simply drag it towards the building or use the start button.
You need to have the ingredients to be able to start production. The necessary ingredients for each recipe are shown on the left side of the screen.
Once the time is up, collect the food from the barn, and it will go automatically to the manger, where the animals will find it by themselves. No need to do anything else!
NOTE: Once an animal has consumed all the available food, it will stop making products until it is fed again. If you don't have the resources to produce food for animals, you can stop feeding them for a while. Nothing will happen to the animals in this case, they will just take a pause in production for some time. Meanwhile, you will be able to focus on more urgent tasks.