Leveling Up and Earning New Kennel Space
As you merge and discover new puppies, you will earn player experience, indicated by the yellow bar at the top left corner of the screen. Earn enough experience and you will level up!
Whenever you level up, you will unlock one more kennel to hold more puppies. So, you should try to be merging as high as possible to get more experience to unlock more puppies.
A good thing to do is to simply leave the game open while you are doing other things. Check back every now and then when your kennel is full, merge to clean up some space and keep bringing in more puppies.