Especially English learning classes for non native English speakers in schools. Google has only given me how it developed in general, and won't tell me how it integrated into schools.
One of the reasons you're likely running into a problem finding an answer via Google is because it's a misnomer to talk about America public schools. That is, due to the 10th Amendment and the lack of explicit mention of education in the Constitution, public education is a matter left up to the states. As a result, every topic associated with public education has 50 different histories. While there are similarities across regions, the role of language classes is idiosyncratic based on population size and immigration patterns. Additionally, America doesn't have an official language, so there has been no federal pressure on states or districts to have English-only instruction but local school leaders could make the call as they saw fit about the languages spoken within the school walls. So, the best answer for your question is going to be tied up in what language and what part of the country in which you're interested. If you have something in particular in mind, I'm happy to answer to the best of my ability!
Meanwhile, I looked at the evolution of non-English instruction here and patterns around immigrants actions after leaving Ellis Island here.