Meditation suffers from a lack of clear and simple understanding for what it is exactly you’re trying to achieve with the practice. With my (limited) experience in reading books on the topic, doing an 8-day silent meditation retreat, and becoming an almost-daily meditator, here is the purpose of meditation as I have come to understand it (so far):
The Buddha was essentially a hacker who, like Neo in The Matrix, hacked the program — known as the mind — that controls the illusion of our universe. He figured out that our thoughts are the root of all suffering. That’s because when you’re thinking:
you’re likely regretting a past event that no longer exists, can’t be changed, and of which your memory is inaccurate anyway, or…
you’re anxious about a future event that also does not (yet) exist, and likely won’t exist as you imagine anyway since the future is uncertain, or…
perhaps you’re ruminating on an unpleasant relationship with someone, imagining what they’re thinking about you when in fact they’re probably not thinking about you at all since people are too self-involved, and so this entire drama is unfolding entirely within the confines of your mind, or…
you’re swept up in a shortlist of other thoughts, all just as equally useless.
Thus, the suffering caused by our minds is needless, since thoughts are empty and meaningless (because they only exist in our head). Therefore, if one could train their mind to control their thoughts, they could end their suffering.
And that is the purpose of meditation: an exercise in mind control that, when practiced consistently over time, gradually conditions you to stave off your unproductive thought patterns. You do this by directing your attention to the present moment, the here-and-now, what is actually real before you — during meditation most commonly the breath (or another “anchor” of your preference).
As you focus on your anchor, you observe the constant influx of thoughts rushing into your mind; they’ll never stop — your mind is always latching onto something — so there’s no need to try and not think. But what you do is notice the thoughts and recognize them for what they are: just thoughts. (Contrast this from what normally happens, which is that you get absorbed by your thoughts and become unaware that you’re just thinking and disconnected from the present moment.) You simply recognize the thought, perhaps categorize it (“past thought”, “future thought”, “sound in the room”, “sensation on the skin”, etc.), and gently return to your breath or other anchor.
When practiced over the long-term, you begin to grow intimately familiar with the workings of your mind (what are the most persistent thoughts running through your head, etc.). You get better and better at immediately catching yourself in a thought before you’ve gotten consumed by it — even when not meditating — and so you can easily redirect your focus to the present moment (your driving, the music, whatever’s happening now), since you understand that to get seduced by the temptation to engage in the drama of your thought is an empty and useless endeavor that can only lead to suffering.
According to the Buddha, the different steps to enlightenment are achieved by this practice. My understanding is the first step is when you’ve practiced long enough to truly internalize the idea that whatever thought pops into your mind is nothing but a thought without any effort, and I hear that this early step is totally achievable by even a casual meditator over time (though mileage may vary).
The reason this makes you calmer, more at peace, less reactive, etc. is because the thoughts that cause emotions, expectations, anxiety, etc. no longer have control over you, and so — like Neo casually observing with detached curiosity the bullets flying past him at the end of The Matrix — you simply disengage from the threat of thoughts, rendering them impotent.
Plus, the present moment is a happy place, filled with excitement and wonder and discovery if you pay attention. Try, for instance, taking the time to savor every forkful of your yummy lunch, rather than scarfing it down while your mind is elsewhere. Now you’re savoring life.
Interestingly enough, one of the later (perhaps the last?) steps to enlightenment is the realization that your sense of self, or ego, is also an illusion — that instead “you”, like everything, are just part of the universe and one of its infinite ways of expressing itself. I’m guessing this level of enlightenment usually can only be achieved — if at all — by a monastic who dedicates their every waking hour to meditation and all forms of “mindfulness” (i.e., staying present with whatever it is you’re doing and what’s happening around you). That or an intense psychedelic experience.
I have found the practice of meditation to be directly applicable to every area of my life. Not only am I quicker to stop myself from getting involved in a thought pattern — a boon for an overthinker like me — but the patience and focus it takes to maintain attention on something as mundane and monotonous as the breath has strengthened my discipline in all things I wish to achieve.