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This is specific to holidays that are celebrated by public schools where only one religion is represented, i.e. Easter, Christmas, etc.
Are public schools supposed to be a neutral ground when it comes to school holidays? Also, does separation of church and state apply at the state level in the US?
There's a difference depending on what you mean by "celebrate".
If you mean that the school is off on religious holidays:
This is, in 2018, largely for non-religious reasons. The schools are out for religious holidays to match the families' calendar, by allowing a large portion of students to not miss day of classes.
As an example, in locales with heavy concentration of religious Jews, public schools have days off on some Jewish holidays. Why? Because lots of kids would stay home anyway; and it's better to just have a day off and not force the students to miss days of classes (theoretically, because it's not good for the kids education. Practically, in reality, it's because the schools get funded per student per day - which means any day a student doesn't attend the school, the school looses funding).
If you mean long vacation timed to a religious holiday (usually "Christmas"/"Easter" vacations), that's to accommodate parents' timing of vacations. Many people take family vacations around that time, and school needs a winter/spring break anyway; so timing it around Easter/Christmas makes practical sense for parents.
If you mean "celebrate" as in "religious ceremonies in the school", that does not happen. Anytime a school does that, someone sues (often ACLU) and recently, the courts have heavily sided with the "no prayer in school" interpretation of the 1st Amendment.
As Brythan's answer noted, this is due to the fact that an official religious ceremony of a specific religion can be seen as a violation of "not establishing a state religion".
If you mean "celebrate" as in have a holiday celebration of non-religious nature, that does actually happen, e.g. for "Holiday concerts" (the artist formerly known as Christmas concert). In more diverse areas, they make extra special effort to have kids do songs about Hanukkah and Kwanzaa, not just Christmas songs. This way, the school isn't "establishing" a religion and treats religions equally.
Technically speaking, the United States doesn't have a requirement for separation of church and state. What it has is a prohibition against the establishment of a religion by the government or restrictions on the free exercise of religion. This is from the first amendment:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
There are some people who believe that the first amendment's prohibition of the establishment of a religion or restrictions on the free exercise of religion is the same thing as separation of church and state.
States also have to follow the rule since the 14th amendment has been interpreted to incorporate the Bill of Rights (the first ten amendments, including the first).
Government operated schools may not require students to join a religion to get schooling.
There are some number of religious universities that receive incidental public funds (student grants and loans) that do have an established religion.
Certain religious people feel that secular schools overly restrict the religious faith of religious students. This remains controversial, as both sides believe that their side has the right of it. One side argues that schools are part of the state and so must be separated from the church; the other side argues that the US constitution prohibits restrictions on the free exercise of religion and the establishment of a "secular church".
Schools in the United States do not have school on the Christian holidays (primarily Easter and Christmas). They provide accommodations for holidays of other faiths (i.e. students may ask to excused on other days and won't be considered truant if observing their religion's holidays). Schools have off both the Jewish and Christian sabbaths but not the Muslim sabbath. Part of this may be that the Jews and Christians explicitly regard the sabbath as a day of rest while Muslims don't.
Public schools generally observe official federal, state, and local holidays. Christmas is a federal holiday, on which basically all government offices are closed. Christmas as a federal holiday is more of a secular than religious observance.
As far as I know, Easter is only a public holiday in Puerto Rico and the US Virgin Islands. I am not aware of any public school in the mainland US that specifically observes "Easter" (though, as it is a Sunday, school is almost never in session anyway). Schools may have a "spring break" some time in March or April, which might coincide with the Easter period in some years, but it is not specifically related to Easter.
