The constitution delegates the power of selecting Electors to the states themselves, through whatever method the state legislature so chooses (today it is always through a popular vote, in most cases statewide but in two cases it is done also by district). Elections for the Senate and Congress are also specifically delegated as to be done however each state legislature so chooses. A right to vote isn't mentioned in the constitution until the suffrage amendments. These protected the right to vote independent of race, servitude, and gender. Moreover, they granted Congress the authority to legislate so as to achieve these ends. So there is federal power intersecting with voting, at least.
Those probably aren't the bone of contention here, however. The claimed issue, as I understand it, is actually from the first part. The claim is that what is going on—extending deadlines etc. via decisions of election boards—is a violation of the Constitution delegating control over voting to the legislatures. As the legislature in question will have of necessity passed legislation empowering the board (or whatever else) to make such a decision on their own, it boils down to whether or not the legislature is held to have the power to delegate that power or not. There's no hard and fast rule for this: sometimes the US Congress has been held to have delegated too much, other times it's held to be just fine (and Kavanaugh, I think, seems inclined to think it has delegated too much in general and this needs to be addressed).
Either way, it becomes a constitutional issue: do actions of election boards, duly enacted by legislation, nevertheless (sometimes) run aground of the Constitutional grant of this power to the state legislatures? Chief Justice Roberts appears to be trying to toe a line where he does not want to get involved with second-guessing state election boards and state courts, wanting to just let those stand as state matters reserved to them by the constitution. His problem, it would seem, is with what federal courts can do, and he seems open to reviewing decisions from federal courts that concern the exercise of state powers on elections. Of course, there are now 5 other "conservative" justices besides Roberts, only 4 of which are needed to hear a case, so the ability for Roberts to get the court to go along with this apparent line of legal reasoning will be strongly dependent on his ability to convince at least two of those conservatives of the correctness, or at least wisdom, of doing so. But that's another matter that goes beyond the question you've asked.