All I can find is that it started as state-legislation in Massachusetts and that it became compulsory in every state by 1918. But how did it become publicly funded ("free") in all 50 states? What was that legislative process?
One of the features of American education history that makes it so important/frustrating to study and research is the disparate school experiences for Black and white children. 1918 is a powerful example of that. While Mississippi was the last state to pass a compulsory education law in 1917, state lawmakers repealed it around the time of Brown v. Board of Education and didn't put it back on the books until 1977. Lawmakers have a variety of motivations when passing or repealing laws, many unknowable to us looking backwards, but the decision to revoke compulsory education was clearly about keeping Black children from accessing white public schools. So as we look back on school taxes, know the focus was primarily white children, mostly boys.
The first law related to education in the colonies is generally recognized as the Old Deluder Satan Law, passed in 1647. In effect, it required households to ensure all children were literate enough to read the scripture, seen as an essential part of being a contributing member of society. In 1651, the law was updated to require towns to provide a school to help households ensure their children were literate. If parents neglected to follow the law, the town would be required to pay 5 pounds to the support and upkeep of a school. It would be misleading to call this fine a tax (and there's not a lot of evidence that it was every actually paid) but it speaks to the relationship between shared responsibility, public funds, and education from the very beginning.
One of the growing pains of the new country included negotiating tensions around the relationship between the government and a child's education. School was generally seen as an add-on to support parents, not the primary source of a child's education. And while the topic of children's education was not seen a pressing national issue, Thomas Jefferson was a vocal advocate for government-funded education. Education in colonial America was a hodge-podge of private, religious, charity schools, private tutors, and Dame schools held in people's homes and Jefferson spoke to flaws in this ad hoc system. His central argument was that well-educated citizens (by which he meant white men) would be inoculated against tyranny and able to defend democracy. In 1779, he wrote the bill For The More General Diffusion of Knowledge, which included a mention of public funds:
To each of the said grammar schools shall be allowed out of the public treasury, the sum of ____ pounds, out of which shall be paid by the Treasurer, on warrant from the Auditors, to the proprietors or tenants of the lands located, the value of their several interests as fixed by the jury, and the balance thereof shall be delivered to the said overseers to defray the expence of the said buildings.
The bill didn't pass. Jefferson was undeterred and wrote several pieces detailing the importance of public education, eventually starting his own college. Around the same time, Noah Webster was coming at childhood education from a different direction through his dictionaries and schoolbooks. His writings echoed Jefferson and raised the public's awareness around the benefits of so-called government schooling. (Webster had a whole bunch of ideas on girls' education if you're curious.)
The government's eventually accepted involvement in education can be seen in the 1787 passage of the Northwest Ordinance. Informed by Jefferson's writing in a similar ordinance of 1784, the document laid out the government's expectations for new states about to join the union and included the phrase: "schools and the means of education shall forever be encouraged." However, as with many things, the devil of "encourage" is in the details and those details are mostly about funding.
Similar to how the education itself was a hodge-podge, funding sources for schools - even tuition free ones open to all students - were diverse. Pennsylvania, for example, used the general state revenue to fund tuition for the children of adults who declared themselves to be a pauper. Maine and New Hampshire used a mix of land grants and poll taxes. A number of schools would use rate-bills, basically tuition fees for parents to fill in the gaps if other means fell short. These disparate systems chugged along for a few generations until Horace Mann arrived on the scene in the 1840's.
The first Secretary of the Massachusetts State Board of Education, Mann was deeply committed to the idea that all American children (which at the time still mostly meant boys, and sometimes girls) would have a common, shared school experience. He was the flag-bearer for the Common School movement which established the formal foundation for school districts and schools as we think of them today. Before Mann there had been a number of attempts at passing tax proposals that failed as working class voters feared they would end up paying for schools that would only serve middle- and upper-middle class students. Wealthier men voted them down because didn't see any reason why they should pay twice: tuition to a private school for their child and in taxes for other people's children. (As historian Joanne Freeman often says, history doesn't repeat but it does echo.)
Mann (and others) tried a different approach better suited for the time. He surveyed schools and submitted reports on the terrible conditions children had to endure just to go an education. He partnered with women like Catharine Beecher to recruit white women to join the teaching ranks (which helped drive down the cost of a school as a female teacher could be paid much less than a male one). He gave well-attended speeches where he orated about the "sacrifice theory" of taxation over "benefit theory", telling the audience that it was their duty to support each other's children for the good of the everyone. He also avoided issues, such as school integration, he thought would hurt his cause. (It likely wasn't a hard decision for him. Like most of his contemporaries, Mann incorrectly believed Black children were less intelligent than white children.)
In years leading up to the Civil War, there was a major evolution in public education from the concept of tuition (funds attached to one child) to taxes (funds from the community to benefit all children in the community). This time around, tax-funded schools were embraced by the working class and seen as a way to improve their children's fortunes. Men of means were now persuaded to view taxes as an investment in the future. Due to the restrictions of the 10th Amendment of the constitution, this all happened on a state by state basis, instead of at a federal level but overall, states settled into a system of local property taxes combined with funds from the state's general tax fund. For new states, these provisions were typically included in their original constitution. Nebraska, for example, included multiple mentions of education funding when they joined the union in 1866. Pre-existing states updated their constitutions through amendments or adopted educational finance laws. By 1890, local property taxes accounted for 68% of all public education revenues in the country.
A great deal would happen after the Civil War, including the creation of an education system for recently freed enslaved people, the assimilation of millions of immigrants, the so-called Great School Wars in New York City which gave rise to the Catholic School system, and eventually greater federal funding through the Elementary and Secondary Education Act of 1965. But in effect, funding that started out as paying for "my" child, evolved into a system for "our" children.