We don't know how Meditations was originally preserved for posterity. The introduction to the Loeb edition (eds. Capps, Page and Rouse) mentions Marcus Aurelius' son-in-law and general Pompeianus, and the emperor's life-long friend Victorinus as individuals who might have 'rescued' the manuscript after Marcus' death in 180 AD. It was also known to the emperor Gordian I (died 238 AD) who was a young man when Marcus died. After this,
The first direct mention of the work is about 350 A.D. in the Orations
of the pagan philosopher Themistius, who speaks of the παραγγέλματα
(precepts) of Marcus. Then for 550 years we lose sight of the book
entirely, until, about 900, the compiler of the dictionary, which goes
by the name of Suidas, reveals the existence of a MS of it by making
some thirty quotations, taken from books I, III, IV, V, IX, and XI.
He calls the book (συγγραφή) an "αγωγή (a directing)
of his own life by Marcus the Emperor in twelve books." About the
same time Arethas, a Cappadocian bishop, writing to his metropolitan,
speaks of the scarcity of this μεγλωφελέστατον βιβλίον, and
apparently sends him a copy of it.
The Wikipedia entry on Arethas expands on the bishop's role in preserving Meditations:
Arethas admits to holding the work in high regard in letters to the
Byzantine emperor Leo VI the Wise and in his comments to Lucian and
Dio Chrysostom'. Arethas is credited with reintroducing the
Meditations to public discourse.
Nothing more is heard or seen of Marcus' work until about 1150 AD when Tzetzes, a grammarian of Constantinople quotes it. Then,
About 150 years later (1300 A.D.) the ecclesiastical historian,
Nicephorus Callistus (iii. 31) writes that Marcus "composed a book of
instruction for his son, full of universal...
experience and wisdom." About this very time Planudes, a monk of
Constantinople, may have been engaged in compiling the anthology of
extracts from various authors, including Marcus and Aelian, which has
come down to us in twenty-five or more MSS dating from the fourteenth
to the sixteenth century. They are practically of no help in
re-establishing the text, and contain in all forty-four extracts
from books IV.-XII.
Our present text is based almost entirely upon two MSS, the Codex
Palatinus (P) first printed in 1558 by Xylander but now lost, which
contains the whole work, and the Codex Vaticanus 1 950 (A) from which
about forty-two lines have dropped out by accidental omissions here
and there.
According to P. A. Brunt in Marcus Aurelius in His Meditations, Wilhelm Xylander's was the first printed edition (the date given here is 1559). Xylander's source was apparently a codex provided to him by the alchemist, doctor and poet Michael Toxites; this source was lost sometime before 1568.

Xylander's Latin translation (from Greek), published by Conrad Gessner
The first English translation was by Meric Casaubon, an French-English classical scholar, published in 1634.