The stay looks to be an administrative one, pending further action of the Court. Witht 4 Justices strongly in favor of disclosure on the Court, I would guess that this results in an eventual denial of the stay, either by a majority vote or a 4-4 split (which keeps the lower court decision in place), unless this is held long enough for Judge Kavanaugh to join the Court.
See this extensive thread from Chris Geidner.
BREAKING: Chief Justice John Roberts, without referring the matter to the rest of the court, issued a stay of a DC Dist Ct ruling that would have meant add'l disclosure requirements on Crossroads GPS and other similar groups. #SCOTUS pic.twitter.com/4gh0SP3rju
— Chris Geidner (@chrisgeidner) September 15, 2018