They don't, and that's actually the point of the FEC's decision. Basically, the FEC has said that determination of eligibility is beyond their purview, and that their authorization only goes as far as enforcing election law as written. Therefore, regardless of whether or not any particular candidate is actually eligible, as long as they follow the FEC's rules, they're fine by the FEC. Nothing more and nothing less.
Then they are an unConstitutional entity and should be abolished.
It is beyond the purview of none but those who are lawless. If that was their ruling they should all resign. Disband the FEC.