Maybe it's that we lack language to articulate what are in fact perfectly bioelectrochemical experiences, due to millennia of religious dominance? And so when we need to describe these inner states, we resort to the language of spirituality?

From this perspective, "what's up with spirituality" is another way of saying "what's up with having feelings, especially important and meaningful and hard-to-explain feelings" and "why are people into exploring their feelings?" Which is totally a valid question.

But automatically couching it in terms of "what's up with spirituality" seems, to me, to be a symptom of the history of language and politics. Are you more interested in the feelings, the framework, or the culturally-specific ways in which people connect the two?

I think you want John Vervaeke's series "Awakening from the Meaning Crisis". Very grounded in the scientific materialist framework, and thoroughly answers your question while also giving a wonderful historical overview of Western meaning-making. You'll know if this is for you after watching the first two episodes, possibly after the first one.

From what I've read, the hormone Oxytocin appears to be behind many of the emotions people generally describe as "spiritual". While the hormone is still being studied, there is evidence that indicates it can increase feelings of connection to entities larger than the self, increase feelings of love and trust with others, and promote feelings of belonging in groups.

The emotion of elevation, which appears to be linked to oxytocin, is most often caused by witnessing other people do altruistic or morally agreeable actions. This may explain the tendency for many religions and spiritual groups to to encourage individuals to share personal or mythological stories that promote elevation. I suspect that the sensation described by many religious groups as "the Holy Spirit" may often be elevation, or at least include elevation in the emotional cocktail. (Some studies indicate that Oxytocin promotes the release of Seratonin, which could be behind the more general feelings of well-being that aren't directly associated with Oxytocin.)

I'm no neurologist or psychologist, so take all of this with a grain of salt. I think the most productive outcome from reading this post would probably be to simply look up Oxytocin and read the studies themselves. 

One of the standard stories is that it's about social cohesion. Especially with rituals done as a group, and other features like visibly taking on costly restrictions in a way that demonstrates buy-in.

Sosis & Alcorta (2003).  Signaling, solidarity, and the sacred: The evolution of religious behavior. 

Different comments here provide different parts of the puzzle: cooperation (especially in large groups) is why, oxytocine (probably) is how. Addressing the facts about spirituality:

Certain drugs seem to induce spiritual or adjacent states

Well, of course, if something is implemented in the brain, there is probably a way to hack the signals by applying chemistry or electricity to the relevant parts of the brain. This is just as true for spirituality as for everything else the human brain does.

People across religions have ‘spiritual experiences’ that involve particular styles of feeling

Yep, people across religions have the same biology.

Spiritual attitudes involve less interest in ‘worldly’ or petty things
Spiritual feelings are often associated with awe and the word ‘transcendent’
Spiritual experiences do not present themselves as about hedonic enjoyment of the experience
Spirituality is related to ideas of connecting with something beyond yourself

These are all about the ability to prioritize something above my personal desires. If I feel that X is more important than my personal desires, and you feel that X is more important than your personal desires, we have a wonderful foundation for cooperation on X-related things! And it scales easily; if you can convince hundred, or thousand, or even million people to prioritize the same X, you get massive cooperation.

There are other possible foundations for cooperation, for example understanding that cooperation can bring mutual benefits. But this is way more complicated! First you need to establish that the cooperation on given project will actually bring more benefits than whatever is everyone's next best alternative. Then you need to agree how to share the profit. Then you need to keep watching out for people trying to stab you in the back and take all the profit for themselves, either individually, or as "inner circles". -- How much easier, if everyone simply believed that X is more important than their profit. Then none of this would be necessary.

And before anyone says "but us nonbelievers can also cooperate: we have countries, armies, corporations, free software, friends helping each other, etc., all without religion", most of these examples either involve some coercion or they don't scale or they have a "religious vibe". You pay taxes because otherwise they would put you in prison; you obey your boss because otherwise you would starve; your friends are limited by the Dunbar number; and your Star Wars fandom is considered creepy by mainstream people.

Spirituality is connected with ‘meaning’ and tends to feel deep and significant
Spirituality tends to be fairly positive, in the sense that from a satisfactorily spiritual mental place, one expects the world to seem broadly good or at least compatible with peace

Well, if something is more important than my personal desires, it should feel significant and rewarding.

Spirituality is sincere

Most people are bad liars, so the best way to convince someone that you value Jesus above your life is to actually value Jesus above your life. Then again, most people are hypocrites, so they will skip the parts where you are supposed to sell all your stuff and donate the money to the poor. Yet, if someone in the community actually sells all their stuff and donates the money to the poor in the name of Jesus, they will probably be rewarded by social status. And people probably won't steal money from the church, because that involves too much risk of going to hell. So... it doesn't work perfectly, but it beats the alternatives. People will cut the corners, but they won't oppose their religion directly, and sometimes will do what is required of them.

Oops, I have conflated spirituality with religion here. But I suspect that this "I believe in unspecific higher power that makes me feel good and requires absolutely nothing from me" is a recent invention. More precisely, freedom of religion is a recent invention. In the past, your tribe had a religion and you either followed it or you got killed/exiled; and spirituality is how you internalized it.

(I wonder if there is any research of how much people who are "spiritual but not religious" are able to cooperate; on the scale from organized religion to atomized atheists.)

Why is it a kind of thing that humans can experience? Why is it a thing they experience so much?

Just to answer this part very narrowly thinking about my own experiences, what is it like when I have experiences I would identify as being the same kind of thing people mean when they talk about spiritual experiences?

The answer is something like a dissolving of self identification into being blended with the rest of the world. In these experiences, I "forget" I'm separate from the rest of the world and am just part of everything. It feels like the subject-object distinction falls away and all is unified.

This can happen to greater or lesser degrees (more or less of the total sum of experience in a moment undergoes this). I suspect this is the kind of thing that would be identified as a spiritual experience.

Honestly, though, I think spirituality is a weird word to use to talk about this and I don't. I just recognize that it seems to be pointing to the same kind of experiences I've described above.

For explanations of popularity of religion, there's identity (simulacrum level 3), the same thing that guides dogmatic political allegiance. It probably has some basic psychological drives behind it, some of which might manifest as spirituality. This predicts spiritual experiences associated with politics.

Spiritual feelings are often associated with awe and the word ‘transcendent’

After being born, at the crucial stages of the development of our being, we are left in a stage of utmost dependency. We own nothing, we are helpless, and the world is full of obvious but unnatainable meaning. At this stage, every growing human needs the percievable world to be fairly well-intended to even survive through infancy, let alone thrive. So evolution hardwired us to seek bonding with our caretakers. That wiring manifests in the form of an intense feeling of fullfillment when we reaffirm or strengthen that bond.

I suspect that transcendence and spiritual awe are more common with people who feel helpless in their life, as an attempt to recreate that feeling of being helped and supported by a powerful entity beyond understanding. Monotheistic spirituality, with the figure of the loving father-god, goes as far as to explicitly state that the believer should forgo understanding and be a helpless child in the hand of the benevolent god.
 

I think you're not asking 'why belief', but rather 'why spiritual high'. The brain produces dmt and the brain produces natural analgesics, we are not sure about the origin of dissociative experiences, but chemicals can induce them. Through certain behaviors, which are given cultural significance, it is possible for some people to get high on themselves.

Removing bullet point 1 from the list is probably appropriate, an organized religion distrustful of spiritual experiences can exist. Some orthodox religions that are out of vogue were explicitly distrustful of these spiritual experiences, and more likely to blame them on satan or djinn than the favor of a venerated deity.

That said if a purveyor of religion figures out a way to create a pleasureable emotional state in others, that state could be associated with the religion or offered as evidence of its' truth for the purpose of persuasion. I think this is common today in the USA.

If you want a shortcut to a spiritual experience without religious context, a look through erowid leads me to think that taking ketamine (nmda antagonist) and DMT should get you there without the need for an angry man in the sky, though you would be breaking some laws and risking real prison time depending on how you got everything together. Therefore, while spiritual experience became associated with religious practice, it probably is not at all.