Pretty much all the books we have from Ancient Rome were preserved by Christian monks. Why did the monks choose to preserve Meditations? Or did we gain it by some other means?
I'm going to give you an in-depth explanation of the manuscript history of the Meditations, as scholars today know it, so be prepared. It's still somewhat under flux, since new scholarship is being done on it as we speak. The TL;DR facts so far:
Current manuscript history
The original Greek text we have and consider to be the Meditations (it's unknown who split it into its 12 chapters, whether it was Marcus or someone else, but even the earliest evidence we have of it split it into 12 chapters, or "books"; we do know that chapters 2 to 12 were written chronologically, while chapter 1 was written after everything else) derives primarily from the transcription of two manuscripts:
These two manuscripts are the only complete ones of the Meditations that we've had access to. They contain some damage and minor missing parts, so the discrepancies between them have been filled in using other, independent fragmentary manuscripts. Today, the text is largely complete other than some occasional missing words. Xylander's printed version was incredibly influential (he was even made the head librarian of the Palatine Library in 1561) and went through several editions, and later translators such as Méric Casaubon (who was the first to translate the Meditations into English in 1643, and as such was also hugely influential) did a lot of correction on the text by using these other fragmentary manuscripts and excerpts.
The Meditations were originally written in Koine Greek by emperor Marcus Aurelius between around 170 and 180 AD, up to 2 years before Marcus Aurelius's death. They were never published by Marcus, as they were strictly personal writings (Marcus refers to them as his "notes" or "notelets" in 3.14.1). For the next 700 years after they're written, we have no surviving record of anyone mentioning the Meditations (some people think that the Greek rhetorician Themistius mentions them in a 364 AD speech, where he says "You do not at all need Marcus’s exhortations", but this is probably a reference to Marcus's wisdom in general, which was well-known throughout history and during his own lifetime, with or without the Meditations).
While Xylander was the first to publicly introduce the Meditations to the Western world in 1558 or 1559, earlier, fragmentary manuscripts mentioning it from the 1300s in the Vatican also exist. In 1517, the German scholar Johann Reuchlin had already made a few quotations of the Meditations in his work On the Art of the Kabbalah. More importantly, it was well-known by the Byzantines at least 600 years earlier:
Oldest source on the Meditations: Arethas of Caesarea (c. 900 AD)
The first concrete evidence for the existence of the Meditations comes from around 900 AD, when Arethas of Caesarea (c. 860–935), a Byzantine bishop and avid book collector, seems to have taken a great interest in the work. He mentions the Meditations in several of his works and letters and indicates that he was a big fan of Marcus. In fact, Arethas is credited by most scholars for having singlehandedly introduced the Meditations to public discourse. So while Arabs and Persians were taking widespread advantage of classical works during medieval times, Byzantine scholars and Western European monks were at least still copying and preserving a lot of classical works. (Interestingly, on an unrelated note, Arethas is the Greek form of the originally Arabic name "Harith".)
So this Byzantine scholar Arethas is therefore the most important figure in the history of the Meditations. He made the first clear, dateable mention of the Meditations of known authorship in a letter he wrote to bishop Demetrius of Eraclea ("Epistols 44") in or before 907 AD. Unfortunately, I don't have access to the text of this letter (L. G. Westerink, Arethae Scripta Minora volume 1, page 305, if anyone wants to go out and save the world), but it generally mentions how Arethas claims to have extracted the text of the Meditations from an old manuscript that reached him by unknown circumstances. (Personal rant: Westerink's goal for publishing "Arethas's minor works" was to finally put them into the world, but no part of his text is available online and the cheapest hardcover version of its is $200 from a sketchy website, and the only library carrying it is the British Library; so much for "putting them into the world".)
Interestingly, Arethas doesn't suggest in this letter that he "discovered" the work; he talks about it as if he merely just copied an existing manuscript in order to better preserve it, indicating that while he is extremely important to modern scholarship on the work and for unwittingly disseminating into a larger audience that allowed it to gain the critical mass it needed to survive afterwards, the manuscript was already at large before Arethas. So it is unknown who preserved the Meditations from 180 to 900 AD, but it was certainly preserved and was relatively well-known in the Byzantine Empire. This unknown (and obviously, also lost) manuscript used by Arethas seems to have been completely undamaged and whole.
Anyways, Arethas is also the source of the only "title" we have for the work, Τὰ εἰς ἑαυτόν ("(thoughts/writings) intended for himself"), which we get from Arethas's work Scholia to Lucian (c. 900 AD), where he mentions it as:
"Pantheia the Smyrnian ... whom the emperor Marcus mentioned in his writings on Ethics intended for himself."
The English title "Meditations" is an even more arbitrary title given by translators. We know absolutely nothing about the work's history before Arethas copied a manuscript he found of it and began mentioning it in his work around 900 AD. And he mentioned it a lot -- including in letters to his friend, the scholarly Byzantine Emperor Leo VI the Wise.
Second oldest: the Suda (c. 990 AD)
After Arethas, the second oldest mention and quoting of the Meditations comes from the Byzantine encyclopedia known as Suda, which did not mention the Meditations before Arethas's lifetime. However, in editions after the late 900s AD, it begins to quote it extensively (a total of 26 times) and makes the famous reference to it as "He wrote a guide for his own life in 12 books" under the entry for Marcus Aurelius.
This is the first work to have actual, direct quotes from the Meditations. While Arethas had paraphrased almost exactly Marcus's words, he did not directly quote him.
P.S.
Another text from the 900s AD that seems to mention the Meditations is the Palatine Anthology (also found in the Palatine Library of Heidelberg University, hence the name), a collection of short Greek poems and part of the larger Greek Anthology. In Book 15 ("Miscellania"), the 23rd poem is this:
On the Book of Marcus
If you want to overcome sorrow, unroll and peruse with care this blessed book, in which you will easily find a great wealth of advice on how you should be, how things are, and how things were. You will see that both joy and pain are nothing but smoke.
While this 1917 translation on Google Books says "nothing is known regarding it", scholarship since 1919 seems to believe that this poem refers to the Meditations (it certainly follows the specific philosophical tone that the Meditations is famous for), and furthermore, that it was added to the Anthology by our very own Arethas himself! However, an interesting hypothesis by Hadot and Luna proposes that this poem may have been written by the earlier Byzantine scholar Theophylact Simocatta circa 600 AD. If that's proven true in the future, it may push back the earliest evidence for the existence of the Meditations from Arethas around 900 AD to this date 300 years earlier.
Sources:
Thank you for the gold, kind stranger.
"It never ceases to amaze me: we all love ourselves more than other people, but care more about their opinion than our own.
If a god appeared to us—or a wise human being, even—and prohibited us from concealing our thoughts or imagining anything without immediately shouting it out, we wouldn’t make it through a single day. That’s how much we value other people’s opinions, instead of our own."
– Marcus Aurelius, Meditations 12.4
were preserved by Christian monks
They were preserved by the Byzantine state and people as well, and remained in the popular culture of that Empire throughout the Middle Ages. The Byzantine Suda encyclopedia/lexicon from the 10th Century writes in the entry for Marcus Aurelius "This man recorded the conduct of his own life in 12 books.", which directly refers to his Meditations. Additionally, a great number of other entries in the Suda provide small quotes from his work.
Many other Byzantine sources from other eras of the Empire's history also make reference to the "philosopher Emperor" and his book. Some Emperors, such as Constantine VII Porphyrogennitos and Romanos III Argyros even tried to emulate him.
Equally interesting is how the work of the other Stoic, Epictetus made its way not just through history but into the hands of the young Marcus Aurelius. I'm not qualified to tell it (though I have published a work on stoicism with a major publisher) but I do urge anyone whose curiosity was piqued by this thread to find that story. It's essentially how the thoughts of a former slave instructed the most powerful man in the world at that time.