I had assumed that Scalia, Thomas, Roberts and Alito were willing to hear the cases and Kennedy blocked them.
It now looks like it was Roberts blocking them and not Kennedy.
Unfortunately, I expect that the full Court will also decline to hear these cases and that this stay will be lifted.
Our best hope now is the Amendment that Senator Cruz is proposing, but I think that's a long shot.
It takes only four votes, not five, to hear a case (but it takes five votes to win a case). Most court-watchers felt that Scalia, Thomas, Alito (and possibly Roberts) voted not to hear the cases because they feared that, if the Court heard the cases, Kennedy would vote with the four liberals to create a right to gay marriage. The conservatives would rather have gay marriage go into effect in 30 states, without a definitive SCOTUS ruling on the issue, than have a definitive ruling imposing gay marriage in all 50 states.
The four liberals could have voted to hear the case, but they perhaps weren't sure that they would get Kennedy's vote (or Roberts'), and preferred to keep the status quo in effect for a while (because public support for gay marriage seems to be growing).
I wouldn't read too much into Kennedy's grant of a stay here; he granted only a one-day stay until he hears from the other side. When he hears from the other side, he will probably refer the stay motion to the full Court. What the Court does then will tell us much more.