The Supreme Court's Brown v. Board of Education decision in 1954 made segregated schools illegal. So when Arkansas Governor Orval Faubus refused to allow nine African American students entrance to Little Rock's Central High School, and mobs of rioters surrounded the kids, Eisenhower had to act to uphold the law and ensure those students' constitutional rights under the equal protection clause of the Fourteenth Amendment.
Executive Order 10730 ordered the Arkansas National Guard to come under the control of the federal government, and ordered the Guard and the Army to restore order and allow the Little Rock Nine to enroll at Central High. Whether they liked it or not, Little Rock was going to admit the students or Eisenhower would send in the troops.
If Eisenhower hadn't put down this obstruction of justice, then the rule of law would mean nothing.
This bottom line in Little Rock wasn't states' rights; it was racism.
In a less-than-classy move, Arkansas refused to enforce federal court orders to integrate public schools in Little Rock, and the governor used the National Guard to keep Black students out of an all-white high school. Bottom line: protestors and Governor Faubus were obstructing justice by making it impossible to carry out the court order.
President Eisenhower ordered protesters against integration to…stop it this minute. When they refused and the National Guard did nothing, Eisenhower ordered the Guard under federal control and authorized the use of the U.S. armed forces to remove the justice-obstructing protestors and enforce the law, i.e., let the students enroll in school.
The president will enforce federal law even if he has to use a lifeline and "phone-a-friend" (a.k.a. the military).