Drop one of the springs (line blasters) down to the wooden cages and swipe out as many of them as you can.
This first step is crucial, look for a column that you can just drop down two blocks to get your line blaster level with a whole row of pets.
Then, carefully drop the other springs down to get the remainder of the pets.
And get them all to the bottom.
Once all these pets are rescued another will drop with your next move (choose a column to click that is easy to get them down)
More pets will drop as others are rescued.