Here's how undocumented students are able to enroll at American universities

In most states, students do not have to disclose their immigration status or provide a social security number when applying for university.

In a landmark judgement, Plyler v. Doe, the Supreme Court decided that the Equal Protections Clause requires local school districts to ensure that all children in the United States have access to K-12 education.

[...] when it comes to higher education three states explicitly bar undocumented students from enrolling in universities:

In those states that have no official policies, undocumented students often must pay out-of-state or even steep international rates for public education.

[...] there are 20 states that not only allow undocumented students to attend institutions of higher education, but also permit those students to pay in-state tuition.

In these 20 states, undocumented youth who graduate from high school within the state and meet other residency requirements – such as having graduated from high school within the state – are eligible to pay in-state tuition in the state’s public universities.

The availability of in-state tuition facilitates access for undocumented college students by making it more affordable.

Some private universities offer a small number of scholarships to undocumented students that enable them to access higher education, but the demand far outpaces supply.

[...] Bard College, which accepts one-third of all applicants, offers much more limited financial aid packages for undocumented students.

Most private universities consider undocumented students to be international students, which often means they have to pay higher tuition than domestic students.

A few, however, have changed their policies and now consider undocumented students to be domestic students, both in their admission criteria and financial aid policies.

Without financial aid, and especially without access to in-state tuition, college attendance remains out of reach for most undocumented students.

To qualify, undocumented immigrants must have been under the age of 31 on or before June 15, 2012; have arrived in the United States before the age of sixteen; and be currently enrolled either in school or in the armed forces or already have completed high school.