One of the many complicated things about American education is that while there is a fair amount that's shared across states and districts, there is no national education system. Which is to say, there are places where a year of formal schooling during a child's 5th year is a policy and cultural norm and places that don't expect a child to start until August/September of their sixth year. Some districts offer fully funded, full-day Kindergarten, and some offer half-day programs. Some have a cultural norm known as "redshirting", where parents delay their child's entrance into Kindergarten to prevent them from being one of the youngest children in the classroom. There is also a wide degree of policy related to a child's date of birth and when they can enroll.

That said, it is fair to say "Kindergarten" is a universal part of the American education landscape, (even before Schwarzenegger’s Kindergarten Cop), leading to a 13 year long school experience for most American children. Generally speaking, the reason Kindergarten became so wide-spread is the idea hit the system and spread right as American education was going through a massive growth spurt which included the rise of educational organizations, laws related to compulsory education and per pupil funding, an increase in the professionalization of teachers and school administrators, the popularity of progressive education, and a national interest in educating immigrant children in the ways of Americana.

First, to its origins. From a previous answer on asking children about their favorite color:

... one particular voice emerged from the crowd of "how to teach small humans" voices and that was a man named Friedrich Fröbel.

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.

The first American Kindergarten class was a small, German-speaking program opened by Margarethe Meyer Schurz in her home in Watertown, Wisconsin in 1856 and the term referred to the teacher, not the students. She was inspired by Fröbel's work in Germany and would soon introduce Elizabeth Peabody, a high profile educator in Boston to the concept. Peabody opened a English-language Kindergarten in 1860 and published her Kindergarten Guide that same year. You can read a later version of it here.) One key feature of Peabody's view of Kindergarten was a heavy focus on the Kindergarten teacher as a mother-figure. That is, she was "gardener" of children. In a practical sense, this meant Peabody encouraged Kindergarten teachers to play with their children, ask open-ended questions, and avoid some of the rigid routines of formal education advocated by her brother-in-law, Horace Mann.

In effect, the women who advocated for Kindergarten in American schools were pushing for something different and new than what had previously been offered. Whereas the structure of common school first borrowed from the classical curriculum and then evolved into a Protestant-shaped liberal arts curriculum, Kindergarten was seen as a progressive space where children would be praised for their curiosity and given structured support to engage with the world around them. We see this sentiment in the work of a Piute woman named Sarah Winnemucca who took the ideas of Kindergarten as envisioned by Peabody and her sister and created the first Indian Boarding School managed by an Indigenous educator, focused on Indigenous education and building Indigenous children's positive sense of self, rather than focusing on assimilation.

For the second half of the 1800s, Kindergarten was mostly likely to be a few hours a day during the year offered by schools, churches, or service organizations across the country by women inspired by Peabody and her writing. She wrote about the power and grace offered by a year of play, sensory experiences, explicit instruction in moving from concrete to abstract concepts, and ways to treat reading and writing as a way for the child to express themselves. One of the reasons her ideas caught on is a different set of educators and school advocates strongly believed that "teaching" young children (i.e. didactic training) was harmful to their development. Peabody was able to thread that needle for four and five-year-olds. In addition, she advocated for the Kindergarten teacher as a bridge between the home and school. Her ideas, sometimes associated with the larger framework known as "social housekeeping" meant the teacher might help with home visits to show parents wants to support their child's learning, and even help connect families with limited resources with social welfare groups.

Two other things to note: whereas teaching was already seen as woman's work (more on that here), Kindergarten teaching was seen as the most womanly of all teaching assignments. This made the work appealing to the families of young woman - mostly white, middle class - who want to earn a living wage. Kindergarten teacher training schools popped up in nearly every American city and given the appeal of Kindergarten teaching as being the ultimate in women's work, were able to be fairly exclusive with their admissions criteria. Which is to say, being a Kindergarten teacher was highly respected among the teaching community as a teacher had to go through a formal teaching program - which wasn't always the case for a teacher. Second, although most education systems up to this point were segregated along racial lines, Kindergarten classes could be found in free and previously enslaved Black communities, following a similar structure to what could be seen in white communities, but not at the same rate. The most well-known of these Kindergartens is likely the Butler School or the Hampton Institute Kindergarten for the 5-year-olds of students, professors, and community members, which likely followed Fröbel's guidance and inspired graduates of the college to open Kindergartens in the cities and towns they moved to after graduation.

The nature of Kindergarten and its role in the system began to shift in the early 1900s. As cities and communities noticed an influx of immigrants, they often formalized Kindergarten within their school system, typically with the thinking that the earlier immigrant children were introduced to American culture and norms, the better off it would be for the child - and the community. At this point, we start to see introductions of more routines and structure and a shift away from Fröbel and Peabody's vision of Kindergarten as a sort of sacred space for children at a particular point in their development. This idea of American culture was very much shaped by white Protestant norms and made Kindergarten look more like elementary school through the introduction of routines like the pledge (more on that here), public performances by the Kindergartners (and a shift of the term from the teacher to the pupils), and an influx of readers and primers into the Kindergarten class. (It is worth stressing that this didn't always mean English-language instruction. Dual language or multi-lingual classrooms were fairly common across America until the 1940s or so.)

The National Kindergarten Association was founded in 1909 and quickly became a leading group among educational organizations. They advocated teachers in elementary school adopt their Froebelian ideas and pulled down ideas related to explicit instruction for five-year-olds related to American culture - routines, songs, theories of Great American Men - and for Kindergarten teachers to be part of teacher unions. Following World War I, German dual language programs typically disappeared. During World War II, as more women entered the workforce, policy related to when a child could start school became codified, and most stand-alone Kindergarten programs ended, or shifted to being a "nursery" or "preschool", serving children under the age of five. By the 1970s or so, even though Fröbel's ideas were mostly forgotten, his term "Kindergarten" wasn't and was the norm across most of the country.