People find puppies so cute because they have the features that generate a visual evolutionary signaling in people to nurture and to care. These features include big eyes, round faces, big round ears, a clumsy walk and floppy limbs.
Many other animals also have these features. The cutest animals that people respond to are animals that also have indicators of "extreme youth, vulnerability, harmlessness and need" reports the New York Times. It turns out that all animals have this cuteness bar, but humans have their cuteness bar set really low so that even animals such as spiders can be considered cute. Puppies are also a companion animal and adding that knowledge with the instinctive cuteness traits make many people respond positively to puppies, which they also consider cute.