Basically, Congress passes laws that grant powers to the Executive to ... execute .. the law and enforce it. These laws instruct the Executive Branch to write administrative policy which is then enforced -- this is the reason, for instance, that the Department of Education could unilaterally rescind the Obama era "Dear Colleague" letter.
The laws that are passed, including Section 230 and the rest of the DMCA, grant the Administration broad powers to write policy from the law in places where the law is insufficiently defined. This is why the Section II/Section III reclassification (called "Net Neutrality" by its proponents, though the actual "Neutrality" is of course controversial) could be done unilaterally by Pai and it's the same reason that the FCC is open to re-interpret the law as written in order to write administrative policy.
If Congress doesn't like the new policy, they can pass a new law that better defines the Executive's role in enforcing their law.