After spending some time in your forest interacting with in-game items, these will appear at random! To catch a petal, you need to pick a butterfly (preferably a smaller speedy one!) and drag this butterfly with your finger into the petal!
The petal and your butterfly will collide, and it will give you a small prize. Once you hit the petal a certain amount of times, you will get a larger prize, and the petal disappears.