In my own (admittedly anecdotal) experience, this curiosity has taken two forms: The intrigue that animals might be more like people than we previously suspected, and the quite distinct intrigue that people might be more like animals. While both are relational, these two types of interest ultimately have different moral centers of gravity.
Non-scientists intrigued by how animal-like people are will often shift the conversation away from animals almost immediately to instead discuss human behavior in evolutionary terms. Dawkins or Pinker might get mentioned. So might some genuinely unfounded evolutionary speculation about mate selection or how tool use arose. At the heart of this curiosity is often the sense that received wisdom about "what makes us human" has it all wrong, and that science is just now starting to peel back the true origins of our most mysterious and remarkable mental attributes.
Contrastingly, non-scientists who are eager to talk about how people-like animals are tend to have a fairly clear idea of what personhood consists of, and are excited by the possibility of extending those properties to animals. They might assert quite confidently that their cat "knows how I feel" or that March of the Penguins was an amazing example of monogamous love conquering all. While this crowd is deeply excited by the journey of self-discovery that life entails, they are much more likely to be convinced they know what the right questions are.
Now, to be fair, I'm generalizing in a rather uncharitable way. Lots of people are interested for lots of reasons. However, as someone who has had to mediate spontaneous discussions-turned-argument between these two camps, I think the two forms of curiosity reflect very different assumptions and require different finesse. It's rather like (and often literally the same as) mediating a discussion about cognition between those who want to overturn the concept of the soul and those who want to prove its existence scientifically. As such, it's very tough to craft a general message that appeals to, excites, and informs both viewpoints.
Nevertheless, the public does seem very excited by these stories, even if the excitement stems from differing motivations. That's excitement that we have to work hard to cultivate, because very few in the general public interesting find animal behavior for its own sake interesting, beyond a sort of "gee whiz" factor that pervades NatDoc film clips in which a species has its entire essence summarized by a 90-second spot. Since good science depends on trying to not fall prey to anthropic assumptions, it's up to us to first hook the general public with the questions that excite them, then try to persuade them that there's a lot to excited about outside of that comfort zone.