Under the language of Obergefell, the Court suggested that traditional Christian values are legally tantamount to discrimination against homosexuals. That requires the Court to carve-out wide religious liberty exceptions for individuals, businesses and NGOs going forward in order to just maintain basic free exercise protections. That is, I believe, what Thomas means by "a problem that only [the Court] can fix."
I don't share the reflexive opposition to any and all conscience exemptions that many other LGBT people have, but it's been established since Reynolds v. United States in the late nineteenth century that the First Amendment protects all religious belief but not all religiously motivated behavior.
The problem the Court has is to determine the extent which private individuals, "closely held" corporations and non-profit institutions can refuse to participate in certain activities under the free exercise clause, and how those claims interact with RFRA. That's an area where the Court has a lot of leeway, and Obergefell pre-empted a more natural progression of state law that would have likely resolved some of these issues absent the intervention of the Court. I think that's all Alito/Thomas are saying here.
