This is from the main opinion:

The Constitution empowers Congress to prescribe how those determinations should be made. The relevant provi- sion is Section 5, which enables Congress, subject of course to judicial review, to pass “appropriate legislation” to “en- force” the Fourteenth Amendment. See City of Boerne v. Flores, 521 U. S. 507, 536 (1997). Or as Senator Howard put it at the time the Amendment was framed, Section 5 “casts upon Congress the responsibility of seeing to it, for the future, that all the sections of the amendment are car- ried out in good faith.” Cong. Globe, 39th Cong., 1st Sess., at 2768

So they say Congress needs to pass legislation to enforce this, and that is the only way to take Trump off the ballot.

The concurring opinion from Sotomayor/Kagen/Jackson does not like this at all:

Although only an individual State’s action is at issue here, the majority opines on which federal actors can enforce Section 3, and how they must do so. The majority announces that a disqualification for insurrection can occur only when Congress enacts a particular kind of legislation pursuant to Section 5 of the Fourteenth Amend- ment. In doing so, the majority shuts the door on other po- tential means of federal enforcement. We cannot join an opinion that decides momentous and difficult issues unnec- essarily, and we therefore concur only in the judgment.