Triple play wrote:
a) abused its discretion;
b1) based its decision on an erroneous legal standard,
b2) or on clearly erroneous findings of fact.
Those might be applicable as rules for the Ninth Circuit in preliminarily staying an order, but I think the SCOTUS basis was different.
The Hilton v. Braunskill standard (quoted by Thomas) codifies:
a) whether the stay applicant has made a strong showing that he is likely to succeed on the merits;
b) whether the applicant will be irreparably injured absent a stay;
c) whether issuance of the stay will substantially injure the other parties interested in the proceeding;
d) and where the public interest lies.
So SCOTUS inter alia found that the stay applicant (Trump) made a strong showing and is "likely to succeed", and there is no substantial injury to parties who have no bona fide link to USA, while there will be irreparable injury against the public interest if Trump's EO is stalled.
https://supreme.justia.com/cases/federal/us/481/770/case.html