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From speaking to chasidim who get married at 18.
They explained this subject like this:

Since it says 18 (if I remember correctly they said this is from the rambam) it is literally understood as the 18th year from birth starting at 0 (0 is 1)
so when someone is legally 17 it is his 18th year and the year he needs to be married (not engaged (in our time the engagement is not engagement just an agreement to get engaged and married at a certain year)).
I do not remember the reason but they said that they get married at 18 but only get engaged at 17


From Bais shmuel 1.3
It seems that only the Bach holds that 18 years means 18 full years (not counting year 0)

The logic behind the number 18
From hilchos Talmud Torah of the Rav 1.1 and 3.1(in the end) is
that since when a person gets married he has the yoke to support his wife and children he will not be able to learn Torah and he has the mitzvah to learn and know the whole Torah. For him to be able to learn the whole Torah he needs to wait until 18 to get married (and maybe it is easier to control your yiester horo before 20), because
at year 5 he starts the written Torah for 5 years,
at year 10 he starts the Mishnah (halocho) for 5 years at
At year 15 he starts the Talmud (the reasons of the halocho) for 5 years
So now he knows the whole Torah
The last 2 year he can do with a wife since it is not so hard (to much time) to support a wife (and children) the first 2 years
At year 20 he needs to go and make a living
See the "Brisa"