Human evolution is the evolution of options. By a wide margin, our species has the most options for responding to challenges. These options come at a great cost, but in the occasional moments when we use them, they catapult our fitness to great heights.
The Chicxulub asteroid filtered animals by temperature. In the immediate aftermath, large reptiles seared to death, whereas animals that could regulate their heat (i.e. "warm-blooded" mammals) burrowed and hid. Not only did they survive the immediate heat of the first couple days, but they survived the climate chaos that ensued. Mammalian metabolic innovations then laid the foundation for high-functioning brain modules due to greater oxygen exchange. These innovations are also shared by birds, who are descended from dinosaurs. While mammals dominate common rankings of the most intelligent species on Earth, those rankings often include a few birds, such as crows, parrots, and owls. There are no reptiles on these lists, though.
Appendages and physical degrees of freedom
In one sense, the primate form is similar to that of octopi. Our appendages are long relative to our bases: long fingers attached to short palms; long arms and legs attached to shorter torsos. Our limbs and digits, when spread out, resemble a nested or fractal hub-and-spoke, meaning that the freedom of our arms multiplies with the freedom of our fingers. The primates are gangly relative to other mammals. If one were to graph animals by the sum of their physical degrees of freedom, it would correlate to their intelligence. Elephants, for example, have the most dexterous of appendages, their trunks. (Dolphins are a great exception, though.)
Human intelligence as depth
The consensus among ordinary people is that humans are the most intelligent species on Earth. However, the world of academia is more interested in countering human hubris, reminding us that, "We're not so special after all." Other species have tool-use, memories, emotions, self-reflection, language, culture, art, creativity, logical thinking, and so on. Every few months a new popular science article reveals something new about animals, whether it's how crows can remember the location of 10,000 objects or how sea lions can deduce transitive logic (if a = b and b = c, then a = d).
If we use basic categories, such as memory, to separate ourselves, then humans are indeed not unique. The rest of the Animal Kingdom has some combination of all the basic pillars of our intelligence. But what's different is our depth. Dolphins have language, but their language can't tell stories. Chimpanzees can use novel tools, but they can't learn a thousand of them like we can.
It's obvious we're smarter than other animals just by looking at the end-points of our endeavors: no other animal writes, does algebra, builds fires, builds ships, or uses computers. No one is contesting the uniqueness of the results of our intelligence. But we don't define human intelligence as fire-building or as computer-usage. So, if we can't delineate human intelligence by its foundations (memories, speech, culture, etc.) nor by its end-points, we have to use something in-between. That middle-ground could be that while all other animals share our basic faculties, they don't have it to the extent that we do. Or as Darwin put it, "The difference in mind between man and the higher animals, great as it is, certainly is one of degree and not of kind."