Your pets have different abilities. Each pet has its own ability (e. g. to fell a tree, a bush, to crush a stone, to collect valuable resources and even explode a dynamite). To use pets' abilities tap on the paw.
Select any pet, which abilities you would like to use. Places, where you can use your pet's ability, will be marked by little paws above. Select any of them and tap on the paw.
To exit the abilities mode, tap on the arrow on the right.
Use all ability charges and wait for the timer to expire.
To recharge pets' abilities, you need to feed them in the Nursery.
Your chosen pet will help you look for additional hidden resources. Don't forget to collect the find from the pet! The find will stay with the pet, until you collect it.