I've answered similar questions before and will be pulling a bit from those answers. First, it's a smidge more accurate to say most students, especially in English-speaking countries, spend 13 years in school - or 12.5 if they do half-day - with Kindergarten.

Societies landed on 12 years through a series of slow changes over the 18th, 19th, and early 20th century - by the 1950s or so, 13 years was pretty much a lock. Rather than emerging as one continuous structure, it was more about the merger of three different structures: grammar school (8 years), high school/secondary (4 years), and kindergarten (1 year). I'll defer to those who know other countries' histories and can best speak to America (which is a bit of a misnomer and there is no national education structure in the US - each state has its own history. That said, there are number of structures and routines that are sharing among the states' and territories' education systems, including the 13/12.5 structure.)

The 8 years of grammar school emerged around the same time as the 4 years of high school. Why 8? It's hard to say exactly why but it's helpful to understand that children didn't enter and leave school based on their ages in the same way they do now. Which is to say, parents started sending their child to school when they felt their child was old enough to go and the child left school when they - or their parents - felt school was no longer necessary. 8 years emerged as the most common structure. That said, grammar school for most of the 19th and parts of the 20th was six to eight weeks in the summer and winter. Younger children were more likely to attend in the summer and older children were more likely to attend in the winter. Beginning on the east coast in the 1830s or so, these two sessions merged into one continuous school year that typically began around Labor Day and ended in late June. (More on that history here under my previous username.)

High school developed as a construct during the same period. The structure went by a few different names: academies, Latin schools, secondary schools - but were intended for students (mostly white, mostly boys) who were either looking for a formal acknowledgement of their studious to show an employer or as preparation for college. During this period, high schools weren't just for teenagers. Students as young as 9 or 10 attended college and needed to master the content necessary for admissions. (Though, not all colleges/universities had admission criteria. Some took any boy or man who was willing and able to pay tuition.) For some young men, high school was their first experience with formal education as their previous experiences were with a tutor. More here, if you're interested, on the evolution of Freshman, Sophomore, Junior, and Senior monikers.

As grammar schools became the norm across the country, a high school diploma was often accepted for employment as a teacher, which meant high schools became increasingly mixed gender as well as less focused on college admission criteria. The count of high schools increased dramatically at the end of 1800s and most high schools settled into a comprehensive liberal arts program that included ancient and modern languages, English literature, math, history, sciences, some form of physical education, and often a gendered course such as homemaking or shop. This shift in focus for most high schools - from something only a few children to something more children did - made it more likely a child would continue on to secondary school from grammar school. This extended the typical school experience from 8 years to 12.

Kindergarten came about in the early 1900s based on the work of Friedrich Fröbel. From a longer response of mine on the history of Kindergarten in America.

Fröbel, a German educator in the early 1800s, is generally recognized as having added two things to pedagogical discourse: Kindergarten and education toys. Without getting too far into his history, he created a series of "gifts" for young children. The first set of gifts, and what is most relevant to our interest was a set of six balls; the three primary colors - red, yellow, and blue and the three secondary colors - purple, green, and orange. The "gifts" and his notions around teaching, known as the Froebelian principles, made their way to America in the late 1800s and early 1900s.

So there you go! Over the course of about 200 years, it became the norm for children to attend school from approximately the age of 5 to 17.