I believe that Konrad Lorenz was the first one to advance the evolutionary theory of cuteness. Stephen Jay Gould wrote an article about it (pdf) using Mickey Mouse as an example (don't be dissuaded by the author's identity). Lorenz argued that we respond with awwwws and nurturing behavior to features that distinguish infant humans from adults, like large round heads, large eyes, small pudgy limbs, and clumsy movements, even if they belong to another animal or a nonliving thing.
There has been research on why animals are cute, again going back to Lorenz, and I think it's generally accepted that the young of many species are cute to us because they share cute-inducing features that are common to that developmental stage, some species (like pandas) seem cute just because out of the wide variety of species some of them happen to have cute-inducing features, and some species (like dogs) seem cute because humans have bred them that way. I'm not sure if there is research on whether adults of other species find their own young to be cute.
Human infants are unusually helpless for an unusually long period of time, which helps explain why humans are so attuned to cuteness (and why there would be a bias towards over-identifying instances of cuteness, which evolutionarily is the less costly error). That doesn't explain why bunnies are cuter than babies, though, or why non-humans dominate the top of our cutest list. Perhaps they just have more of the cute-inducing features. Humans occupy a small portion of body-space, and if you move from the region occupied by adult humans to the region occupied by human infants and then go even further along the same dimensions, you could run into regions occupied by other animals. But why would this happen for cuteness but not sexiness?
This is a rewrite of my comment as more of an argument and less links and speculation, since I think that parts of it might be clearer that way.
Lorenz's theory is that humans evolved to respond with an awwww to the features that distinguish infants from adults, and so we also awwww to other animals that have those features. Why do other species have features that we find cute? One reason is that we've exerted selection pressure on them - for instance, by being more friendly to cuter wolf/dogs. A second is that features common among mammal young naturall...
Daniel Dennett has advanced the opinion that the evolutionary purpose of the cuteness response in humans is to make us respond positively to babies. This does seem plausible. Babies are pretty cute, after all. It's a tempting explanation.
Here is one of the cutest baby pictures I found on a Google search.
And this is a bunny.
Correct me if I'm wrong, but the bunny is about 75,119 times cuter than the baby.
Now, bunnies are not evolutionarily important for humans to like and want to nurture. In fact, bunnies are edible. By rights, my evolutionary response to the bunny should be "mmm, needs a sprig of rosemary and thirty minutes on a spit". But instead, that bunny - and not the baby or any other baby I've seen - strikes the epicenter of my cuteness response, and being more baby-like along any dimension would not improve the bunny. It would not look better bald. It would not be improved with little round humanlike ears. It would not be more precious with thumbs, easier to love if it had no tail, more adorable if it were enlarged to weigh about seven pounds.
If "awwww" is a response designed to make me love human babies and everything else that makes me go "awwww" is a mere side effect of that engineered reaction, it is drastically misaimed. Other responses for which we have similar evolutionary psychology explanations don't seem badly targeted in this way. If they miss their supposed objects at all, at least it's not in most people. (Furries, for instance, exist, but they're not a common variation on human sexual interest - the most generally applicable superstimuli for sexiness look like at-least-superficially healthy, mature humans with prominent human sexual characteristics.) We've invested enough energy into transforming our food landscape that we can happily eat virtual poison, but that's a departure from the ancestral environment - bunnies? All natural, every whisker.1
It is embarrassingly easy to come up with evolutionary psychology stories to explain little segments of data and have it sound good to a surface understanding of how evolution works. Why are babies cute? They have to be, so we'll take care of them. And then someone with a slightly better cause and effect understanding turns it right-side-up, as Dennett has, and then it sounds really clever. You can have this entire conversation without mentioning bunnies (or kittens or jerboas or any other adorable thing). But by excluding those items from a discussion that is, ostensibly, about cuteness, you do not have a hypothesis that actually fits all of the data - only the data that seems relevant to the answer that presents itself immediately.
Evo-psych explanations are tempting even when they're cheaply wrong, because the knowledge you need to construct ones that sound good to the educated is itself not cheap at all. You have to know lots of stuff about what "motivates" evolutionary changes, reject group selection, understand that the brain is just an organ, dispel the illusion of little XML tags attached to objects in the world calling them "cute" or "pretty" or anything else - but you also have to account for a decent proportion of the facts to not be steering completely left of reality.
Humans are frickin' complicated beasties. It's a hard, hard job to model us in a way that says anything useful without contradicting information we have about ourselves. But that's no excuse for abandoning the task. What causes the cuteness response? Why is that bunny so outrageously adorable? Why are babies, well, pretty cute? I don't know - but I'm pretty sure it's not the cheap reason, because evolution doesn't want me to nurture bunnies. Inasmuch as it wants me to react to bunnies, it wants me to eat them, or at least be motivated to keep them away from my salad fixings.
1It is possible that the bunny depicted is a domestic specimen, but it doesn't look like it to me. In any event, I chose it for being a really great example; there are many decidedly wild animals that are also cuter than cute human babies.