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You're right that Star Trek species are generally portrayed as monolithic societies¹. What makes humans special is that they don't have that. Our Hat is not to have a Hat.

Ambassador Soval reveals to Admiral Forrest that this is the opinion of the Vulcan High Command in Enterprise season 4 episode 7, "The Forge"²:

Soval: [The Vulcans] don't know what to do about Humans. Of all the species we've made contact with, yours is the only one we can't define. You have the arrogance of Andorians, the stubborn pride of Tellerites. One moment you're as driven by your emotions as Klingons, and the next you confound us by suddenly embracing logic.

Forrest: I'm sure those qualities are found in every species.

Soval: Not in such confusing abundance.

So the Vulcan opinion is that non-human Star Trek species are, with very few exceptions, as monocultural as they're portrayed on the show.

An alternative opinion is expressed in "The Cage", the original pilot for TOS³. The premise of the episode is that Captain Pike has been taken captive by the Talosian species, who capture and study members of other species. One of the Talosians, referred to as The Keeper, has this to say regarding humans:

The Keeper: The customs and history of your race show a unique hatred of captivity.

[Later]

The Keeper: No other specimen has shown your adaptability.

The Talosians have a similar, but slightly different perspective to the Vulcans: they believe that Humans are uniquely disposed against captivity (although it's not clear how many sentient species they've studied), and are uniquely able to adjust how they respond to situations. Your mileage may vary, but as far as I'm concerned this is another way of saying that the human Hat is to not have a Hat.

As far as I know, there's no in-universe explanation for this.

One possibility is that it's because Earth has been a united planet for so short a time; Memory Alpha tells me that the United Earth government was founded in 2150⁴. We've observed in real life that globalization tends to limit cultural diversity, so it's conceivable that a few hundred years of single-government would eventually reduce us to a Planet of Hats.

¹ TVTropes refers to this as Planet of Hats

² I should point out that it's possible this is just Vulcan prejudice; one of the recurring ideas of Enterprise was that Vulcans tend to look down on species they view as inferior, and they view all species as inferior in some regard. So this may not be the most unbiased view, but it does mesh with what Star Trek shows us of non-human species

³ Although "The Cage" was ultimately rejected, pretty much all of the footage was re-used in the later TOS two-parter "The Menagerie", so this is still canon

⁴ The year before Enterprise begins, about a hundred-ish years before TOS, and about 200 years before TNG and its spin-offs. To contrast, the idea of a unified Vulcan society most likely begins with Surak and the Time of Awakening, sometime around what Earth would consider the year 300

ST:TNG "Hide and Q" has a plausible answer.

There, Q tells Riker that the human desire to explore and to learn, as well as their curiosity, will eventually lead humans to a higher state of being. Q then went on to say that humans might even surpass the Q.

Q: Of course you don't, and you never will until you become one of us.

RIKER: Until? Would you mind going over that again?

Q: Well if you'll stop interrupting me. This is hardly a time to be teaching you the true nature of the universe. However, at Farpoint we saw you as savages only. We discovered instead that you are unusual creatures in your own limited ways. Ways which in time will not be so limited.

RIKER: We're growing. Something about us compels us to learn, explore.

Q: Yes, the human compulsion. And unfortunately for us, it is a power which will grow stronger century after century, aeon after aeon.

RIKER: Aeons. Have you any idea how far we'll advance?

Q: Perhaps in a future that you cannot yet conceive, even beyond us. So you see, we must know more about this human condition. That's why we've selected you, Riker, to become part of the Q, so that you can bring to us this human need and hunger, that we may understand it.

In addition, there's an angle on the uniformity of human culture (as little of it as there exists) in an episode where AmerInd is settled with Native Americans by unknown aliens, to preserve their culture.

I think it is a matter of perspective. For a typical European, all Chinese and Japanese people look very similar, but can probably tell apart a typical Swede from an Italian; and the other way around.

The variability in characters is actually quite similar across species. Some Vulcans are more emotional than others, some Klingons are more prone to distrust and scheming than others... but as they are more different than us, we notice that more than the internal differences.

As no scientific argument is complete without a plot, here is one, depicting humans in blue and klingons in green.

Even though both clouds have the same variability, from anywhere inside the human cloud we see the klingons far away, and in the same general direction. After all, in our daily life, there is little difference in dealing with a person as violent as Gowron, or one closer to Martok; and many people will be equally annoyed at having Spock or T'Pol as managers. But of course, as Klingons or Vulcans go, they are indeed different.

In Star Trek, the stories are usually presented from the perspective of the protagonists, who are usually human, part human, or well-acclimated to living with humans. Being outsiders with respect to alien cultures, they naturally see those alien cultures as somewhat monolithic -- just as Westerners might see a foreign country like China as a mono-culture.

I also agree with the answer that pointed out Star Trek's chronology, and how it begins only a few hundred years after a united Earth government, with other species generally attaining world government much earlier. This naturally leads to more diversity among humans compared with aliens.

Some other answers point out human adaptability, versatility, curiosity, and drive to explore, and those are certainly highlighted in Star Trek as quintessentially human qualities. But I think I would also add two more: morality and self-improvement. In Star Trek the ideal human (e.g. Picard) is presented in his best moments as a seeker and follower of a set of moral guidelines that are close to "universally good", and generally superior to those of other cultures. (Of course, imposing those morals on other cultures is frowned upon by the Prime Directive, even though the protagonists frequently do so with solid justification, making the Prime Directive seem shaky at best -- but that is a whole other question.) Among the other cultures, only Vulcans perhaps possess such standing, and even Spock says that logic is "the beginning of wisdom, not the end," implying that the Vulcan devotion to logic is, on some level, as limiting as the Klingon warrior code of honor. So humans have a unique drive for self-betterment, not limiting themselves to a specific mental or martial code.

Humans are the only superpower race that are shown to have unified and expanded under a civilian government (although seemingly with some help from the Vulcan military).

Most of the other major powers of the quadrant all attained prominence under military leadership and/or spent a great deal of time in such a culture. The Romulans and Klingons have no apparent distinction between the military and executive arms of their government and show no sign of any functioning democracy. Andoria, Vulcan, and Cardassia have spent centuries as de facto military governments overriding the underlying system (although Cardassia at least had a nominal civilian government sharing power with - not leading - the military).

The other Federation races are never developed in much detail, but there's no real indication that e.g. the Betazoids or the Trills, who are peaceful minor powers, are particularly hat-wearing. In contrast, many human colony worlds (as shown in early seasons of TNG) do have strong hats - and strong authoritarian centralized government.

Bajor is explicitly shown in the process of emerging onto the galactic stage in Deep Space Nine under a new civilian government, and its society is shown to be at least as diverse as that of humans (it is one of the few planets portrayed as having more than one region or language, and major issues in Bajoran politics are a focus of the early seasons of DS9). Cardassia is also mentioned - not really shown - as experiencing periodic cultural renaissances whenever it manages to break the cycle of dictatorship for a few years.

So it's possible that humans are unusual in this respect not because of anything specific to the species, but because of the happy combination of a civilian-led emergence and formative period, cemented by gaining powerful allies very early on and avoiding the need to resort to authoritarian, centralized rule in order to survive. Earth managed to avoid the pressures that lead to the development of a hat.

People will tell you that in the galaxy of hats humans are unique for not having a hat. By the time of TNG that's wrong. The human hat as decreed by the Great Bird of the Galaxy is being "nice and friendly".

Quark: I want you to try something for me. Take a sip of this

Garek.: What is it?

Quark: A Human drink, it's called root beer

Garek: I don't know

Quark: Come on aren't you just a little curious

Garek: /sigh (takes a sip) It's vile!

Quaek: I know, it's so bubbly and cloying and happy.

Garek: just like the federation

Quark: but you know what's really frightening? If you drink enough of it, you begin to like it.

Garek: It's insidious

Quark: just like the federation

Of course not all humans are actually nice. But then not all Vulcans are logical. And of course humans are nice until you push them a bit too far...

Quark: Let me tell you something about Hew-mons, Nephew. They're a wonderful, friendly people, as long as their bellies are full and their holosuites are working. But take away their creature comforts, deprive them of food, sleep, sonic showers, put their lives in jeopardy over an extended period of time and those same friendly, intelligent, wonderful people... will become as nasty and as violent as the most bloodthirsty Klingon. You don't believe me? Look at those faces. Look in their eyes.

In most fictious works the human race's main trait is ambition, and Star Trek is no different. From their first warp engine in 2063 (Star Trek: First Contact) and within a century, with the forming of The Federation in 2161 (Star Trek: Enterprise), they had become one of the main players in the solar system. A few years earlier ambassador Soval suggested that the Vulcans were afraid of humans because the same feat took them nearly 1500 years (Star Trek: Enterprise, "The Forge").

As i said, the same keep happening in a lot of other fiction: Mankind arrives late to the scene and quickly they become one of the most important races and insist on meddling in anyone elses affairs. In the Mass Effect series the humans bypassed multiple races to a position on the Council in a very short time. In Lord of the Rings they pushed the elves of Middle Earth with their empire building. And even in Dungeons & Dragons the humans usual trait is that of ambition, and to mirror that in the rules they usually gain an extra feat or skillpoints.