I've also read the Wikipedia entry on the case, and I encourage you to both watch the video (if you haven't already) and read the wiki.
When I first encountered this decision, my gut reaction was positive. While the video is well-made, and the SCOTUS decision a close one (5-4), I'm still inclined to stick with my initial impression. There were a few things I think the video got wrong. What Vanderbuilt is doing is withdrawing access to campus resources for groups that do not conform to the "all comers" policy. It is not dissolving those groups, "kicking them off campus", or taking any administrative action against the members of those groups. In this, I think the video overstates the case against the policy, as it's not clear to me that any university is required by law to provide resources to groups whose restrictive policies do not conform to a set of standards the university feels is acceptable.
The case, to my mind, is a conflict between two fundamental "rights" and the degree to which each should be protected in the context of the other. The first question is whether an individual has the "right" to join any group, whether or not they agree with the groups aims or meet the group's standards of "belief or conduct". The second is to what extent student organizations have a "right" to the resources of the university.
For the first, I think this right is necessarily protected by the university that hosts, e.g. provides aid and comfort to, organizations that would, ostensibly or otherwise, represent the university (even by proxy). So organizations that utilize the campus facilities, print and publish beneath the university's letterhead, or otherwise conduct business under the university's umbrella of pseudo-sponsorship, fall under the university's responsibility. In that sense, any individual attending that university as a paying student has, or in my opinion should have, the "right" to join any group they so choose, regardless of whether they agree with the group's overall aims, or meet the group's non-academic standards of entry. Any organization that discriminates, not just based on status (race or gender), but on "belief or conduct" (religion, sexual preference, ideological or political affiliation), is doing so with the implicit endorsement of the institution that houses it and allows it to represent that institution.
For the second question, to what extent do student organizations have a "right" to the resources of the university, I think this right is predicated on what sorts of messages the university wants to endorse. The freedom of assembly guaranteed by the Bill of Rights extends only insofar as the legal contract between citizens and their government. It does not require that private institutions hold host to, or financially or implicitly support, that assembly. Universities could in principle cease funding or supporting all student organizations, because while they cannot stop the people from assembly, the do not have to host or support them either.
So in reality, the second question, to what extent do student organizations have a "right" to the resources of the university, is predicated on the answer to the first question, the right of an individual to associate with any university endorsed student organization. The university, rather than the student organization, is the entity faced with the choice, and the responsibility, for the practices of the organizations it hosts.
Much of the criticism of this decision, as you can see from the video (that you all watched, I'm sure), stems from the idea that these organizations have a right(not necessarily a legal right) to retain their ideological "integrity". One example is a Christian organization's "right" to exclude non-Christians, or a gay-rights group's "right" to exclude those who do not feel gay's should be allowed the right to, say, get married. Other examples would be the Democratic group, or any political group, that risks having it's message tainted by mixed messages from it's members. These are, of course, valid concerns for any organization, and are why the freedom of assembly is included in the Bill of Rights.
In Roberts v. United States Jaycees, the Supreme Court held that associations may not exclude people for reasons unrelated to the group's expression. However, in the subsequent decisions of Hurley v. Irish-American Gay, Lesbian, and Bisexual Group of Boston, the Court ruled that a group may exclude people from membership if their presence would affect the group's ability to advocate a particular point of view. The government cannot, through the use of anti-discrimination laws, force groups to include a message that they do not wish to convey. (Wikipedia)
Still, it's not clear that these groups have both a right to assemble in that manner, and the right to the resources of the university, concurrently. If the Christian organization wishes to exclude based on creed, and the university they attend is uninterested in hosting such behavior, that organization is still free to exercise it's right to assembly even if the university is unwilling to support it. The same applies to any other type of organization, e.g. a "Male Supremacy" group that is unwilling to allow women to attend. The group has not lost it's rights guaranteed by the Constitution simply because the university is unwilling to support that group, and thus, in my humble opinion, has no meaningful redress under the Constitution.
Why is this not just like the freedom to express thoughts and ideas like those other videos you agree with?
I'm glad you asked! I view the difference in a student's right to express himself or herself on or off campus as protected by the 1st amendment irrespective of his or her status as a student. The university is not required to support said speech, so Vanderbuilt probably shouldn't be required to provide a platform for speech it doesn't agree with. Just as they cannot and should not take administrative action against individuals who join or maintain groups whose exlusionary policies do not reflect the institution's position (and thus are not entitled to the institution's support), individuals who express unpopular views should likewise be free from administrative action. If someone wishes to preach their opinion about the inferiority of black people, or white people, or gays, etc., they are certainly free to do so, but the university is not required to pay for the microphone and lectern from which such a view may emerge.