Pets fall in the bottom section of the game first (none will appear in the top until there is a Pet below)
When you have a Pet in the bottom, choose a column as far to the left as you can to get the next Pet to drop in.
Work on filling the butterfly to save the trapped Pet - be aware that the butterfly will always save the highest Pet in the game. So most often in this game, the one to the left will be saved first because it is generally higher.
The Pet in the top section can also be saved by getting it down to the bottom.
Try to get as many Pets as you can, down and rescued in the first column.
Use the rocket to help move this Pet along (if needed) and down.
If you manage to stop the Pet in the first column lower in the game than the trapped Pet, it will remain while trapped Pet is saved by a butterfly.
Then, and only in this scenario, you are able to get two Pets in the left/top section of the game. Make your next move on this side of the board to get both Pets together.
This means you can get two down with one move (using the in game rocket, or by matches)
(The second play through of this Level in the video example shows this method)
More Pets will not appear until you get another in the lower section of the game again. So make your next move down in it.