Educational opportunities expanded after the Civil War because the government set up more and more public schools. This was done partly out of a desire to "Americanize" the children of the immigrants who were coming into the US in large numbers starting in 1880. In part, though, it was also done because of the idea that educated people were needed for a democratic society to exist.
After 1870 in particular, public education took off. The number of teaching colleges ("normal schools") went from 12 in 1860 to over 300 in 1910. (Source: Bailey, et al. The American Pageant 11th Ed. p. 580.) In 1870, 57% of school age children in the United States went to school. By 1920, even with the influx of immigrants, that number was 78%. (Source: Tindall and Shi, America: A Narrative History 5th Ed. p. 948).
Educational opportunities expanded, then, as governments set up more public schools after the Civil War.