What is the historical reason for 18 years being the most commonly accepted age of adulthood?

There were several reasons for 18 years old being the threshold of adulthood.

The first reason was military, as embodied in the 26th Amendment. During World War II, Congress lowered the draft age from 21 to 18, more or less in line with the other major combatants. This actually became a sore point during the Vietnam War, when the slogan became, "old enough to fight, old enough to vote." And "voting" was basically the key right of adulthood.

A second reason was that until the post war period, most children completed their (secondary) education at the age of 18, and entered the work force at that time. Here, again, they had "adult" responsibilities at age 18.

A third reason is that 18 is the maximum age of "consent" for marriage, and "marital relations" (including those undertaken outside of marriage). Although some states and countries have lower ages of consent.

It's true that for most people, particularly men, intellectual development is not complete, and perhaps one is not fully adult until age 25. (This is why a Congressman must be that age.) But that's not the point, which is that an 18 year old is "more adult than not," which is why the line is set there.

Perhaps it is worth mentioning the Jewish traditions, although it is hard to say to what extent they might have influenced Christian perceptions of age.

Bar Mitzvah (Bat Mitzvah for girls) is a ritual that is celebrated at the age of 13. It is considered as a coming of age, where a boy abandons the society of women and joins the society of men. To succeed in this ritual he has to demonstrate signs of maturity, which includes basic literacy - like reading from Talmud, answering questions about the basics of the Jewish law, etc. This seems particularly meaningful in societies where there is gender separation in regard to prayer, household duties, military obligations, etc. Indeed, in such traditional societies little boys are cared for by their mothers and grandmothers, with little direct participation of men. Yet, at some point this transition must happen, and it requires both some preparation and testing readiness for the adult life.

Reaching the age of bar or bat mitzvah signifies becoming a full-fledged member of the Jewish community with the responsibilities that come with it. These include moral responsibility for one's own actions; eligibility to be called to read from the Torah and lead or participate in a minyan; the right to possess personal property and to legally marry on one's own according to Jewish law; the duty to follow the 613 laws of the Torah and keep the halakha; and the capacity to testify as a witness in a beth din (rabbinical court) case.

The tradition for girls mostly has to do with their readiness for marriage. My understanding is that it climbed in importance due to the development of the reformist/non-orthodox Judaism, tending to emancipate women.

A couple of quotes from Wikipedia, regarding the specific age:

The Bible does not explicitly specify the age of 13. Passages in the books of Exodus and Numbers note the age of majority for army service as twenty. Machzor Vitri notes that Genesis 34:25 refers to Levi as a "man", when a calculation from other verses suggests that Levi was aged 13 at the time.

The age of thirteen is mentioned in the Mishnah as the time one is obligated to observe the Torah's commandments: "At five years old one should study the Scriptures, at ten years for the Mishnah, at 13 for the commandments..."

and

The term "bar mitzvah" appears first in the Talmud, meaning "one who is subject to the law", though it does not refer to age. The term "bar mitzvah", in reference to age, cannot be clearly traced earlier than the 14th century, the older rabbinical term being "gadol" (adult) or "bar 'onshin" (one legally responsible for own misdoings).

On a minor detail, as far as I know ancient Roman law had several stages and ages of majority.

Roman boys had a ceremony to remove the toga praetexta with a broad purple border, and their protective bulla praetexa amulets, and assume the pure white toga virilis, the toga of manhood, at 14, considered to be the age of male puberty in ancient Rome. Roman boys would have their first military service sometime in their late teens. And I believe the end of adolescence and the age of complete adulthood was 25. And the minimum age to run for elected public office was 30.

So the closest fit between ancient Roman and modern legal ages is that the minimum age for military service is similar.

I believe that the main reason why the legal age has been internationally set at 18 years of age was because of WWII. During the Second World War, the military draft age was lowered to 18; 18-year-olds could be legally drafted to fight wars for their nations. The Vietnam War was a pivotal moment; if 18-20-year-olds were being forced to fight for their nation, why is that they were still not allowed to vote?

But if you actually read history, you will see that the age of majority is really meaningless for much of history. You will find that most minimum legal age restrictions were either much lower (12-14 years) or non-existent. Modern society places lots of weight on the legal age; your 18th birthday is a drastic turning point, legally. According to the law, the moment you turn 18, you can sign contracts, work unrestricted, purchase firearms, own property, marry without parental consent, consent to sex with anyone, and so forth. There is a large difference between 17 and 18, legally, even though the biological difference is minimal.

If you look at different time periods though, different societies set different age limits. 20 was the Biblical age of military service, for instance (Numbers 1:1-3).

There is nothing special about 18. The really scary thing is that modern people think that 18-year-olds are children, whereas decades back, they were fully functioning as adults. It was not uncommon to be married by 18 a few decades back.

We are definitely not rushing anyone into adulthood. If anything, childhood has been extended well into the 20s with disastrous effect.

For a long time, 21 was the "age of adulthood". From Encyclopedia of Adult Development:

As early as the thirteenth century, age 21 was an important marker in Britain, at least for men. This was the age, for example, when they could begin to serve as knights. It was believed that the physical requirements of combat - to wear a heavy suit of armor and simultaneously lift a sword or lance - could only be met by men who had attained age 21.

It wasn't until the 26th amendment in 1971 that the voting age in the United States was lowered from 21 to 18. A big reason for this was the outcry that men were "old enough to fight, but too young to vote".

I would guess that the reason men were drafted into the military at age 18 was because of the need for as many soldiers as possible and since armor was no longer in use, a person did not have to be as physically developed to serve as a soldier. Additionally, most men finish high school at age 18 so they have all of their primary and secondary education completed.