What is the history of the expression “many moons ago”?

Many moons ago is an old-fashioned expression that means a long time ago. Moons refer to months (month derives from moon) and the expression meaning is just literal. According to Ngram its usage is from the 18th century. It appears the expression is just a popular/common way to refer to an ancient measure of time.

• The word, "moon" is the origin of the word, "month," which is roughly the amount of time it takes the moon to travel through its all of its phases in the sky. Months on the Julian calendar are named after Roman gods, emperors and numbers. The Algonquin tribes of the Northeast took a more natural approach, and named their full moons (and the month it appeared in) after what was going on around them.

(www.bayjournal.com)

Month:

• Old English monað, from Proto-Germanic *menoth- (cognates: Old Saxon manoth, *related to menon- "moon" (see moon (n.); the month was calculated from lunar phases). Its cognates mean only "month" in the Romance languages, but in Germanic generally continue to do double duty.

(Etymonline)

The English translation of many moons ago is first attributed to the Kuna people, previously known as Cuna Indians, who like many other Native Americans tracked the passage of time based on the full moon cycle.

The following excerpt is from A New Voyage and Description of the Jsthmus of America, first published in 1695 in London; the author Lionel Wafer was a ship surgeon, explorer and buccaneer who was stranded on the Isthmus of Panama. Wikipedia claims the Welshman spent four years, 1680-1684, in Panama, but according to a second source, he spent only four months. Wafer cohabited with the Cuna Indians, and became friends with their king named Lacenta. On his return home, the Welsh ‘pirate’ wrote about their culture, their shamanism and a short vocabulary of their language.

[emphasis in bold mine]

The Indians, when they travel, guide themselves either by the Sun, when it shines, or by steering towards such a determinate Point, observing the bending of the Trees, according as the Wind is. If they are at a loss this way, they notch the Barks of Trees, to see which side is thickest; which is always the South, or the Sunny Side; and their way lies generally through Woods. […]

I observ'd among them no distinction of Weeks, or particular Days, no parting the Day into Hours, or any Portions, otherwise than by this Pointing: And when they use this, or any other Sign, yet they speak at the same time, and express their Meaning in their own Language, tho' to Europeans who understand it not. They reckon Times past by no Revolutions of the Heavenly Bodies, but the Moons: For Lacenta speaking of the Havock the Spaniards had made to the Westward, intimated 'twas a great many Moons ago.

Their Computation is by Unites and Tens, and Scores, to an Hundred; beyond which I have not heard them reckon. To express a Number above this, they take a Lock of their Hair, little or great (in proportion to the Number they would intimate) and hold it up in their Hands, sorting it gradually with their Fingers, and shaking it. To express a thing innumerable, they take up all the Hair on one side of the Head, and shake it.

According to one Native American legend, the Winnebago Legend, the good spirits and the evil spirits agreed that the calendar year should be divided into 12 moons, which corresponded to the number of stripes on one chipmunk's tail. The six white stripes represented the ‘winter moons’ while the remaining six black stripes were the ‘summer moons’. However, as a lunar month typically consists of 29 days, for many Native Americans it meant the year was divided into ‘thirteen moons’.

For some tribes, the year contained 4 seasons and started at a certain season, such as spring or fall. Others counted 5 seasons to a year. Some tribes defined a year as 12 Moons, while others assigned it 13. Certain tribes that used the lunar calendar added an extra Moon every few years, to keep it in sync with the seasons.
Each tribe that did name the full Moons (and/or lunar months) had its own naming preferences. Some would use 12 names for the year while others might use 5, 6, or 7; also, certain names might change the next year. A full Moon name used by one tribe might differ from one used by another tribe for the same time period, or be the same name but represent a different time period.

Source: Almanac.com

I don't know the etymological origin, but to my mind it's a fairly simple historical relic. Historically, knowing the calendar month or date was much, much rarer than we now take for granted.

A typical Medieval farming community would instead count the passage of times by the moon's waxing and waning. In that context, a 'moon' from the expression would be a full cycle of the moon, and so 'many moons ago' would be just as natural to say then as 'several months ago' might be now.