I was more under the impression that they used Trump's campaign and post-election comments as exigent evidence that the true purpose of the ban was to target people based on religion, and therefore in violation of the 1st Amendment. I'm fine with this argument, but not with the fact that they agreed that Washington state had standing to sue.
Whatever, nobody cares about the technical details of law, only that their side "wins" the outcomes.
They didn't get to the point of addressing whether or not the law was targeting a religion at all, just completely punted on it. The temporary stay was upheld solely on the basis the states (Washington & Minnesota) were able to demonstrate that the ban was causing significant harm to them. Businesses were not able to have employees travel, universities were without faculty and students present, families were separated, and various other people with granted legal rights to be in the U.S. weren't allowed in any more. This is costing the states money (i.e. lost tax revenue), and residents of the states hardship. The state universities being harmed here being the most argued point (as far as this case goes) as for why the states have a right to bring legal action.
Part of this was made worse by the vagueness of the executive order. The confusion on whether it applied to people with valid visas, green cards, etc. In a sense because of some on the initial confusion and vague wording of the original document (so are people with valid visas allowed in or not? What about those with dual citizenship?) they were convinced the order could be used to limit access to these people, even if that wasn't necessarily how it was intended, so that argument held ground. Trump's words hurt here as well, of course. Adding to evidence that the government could choose the broader interpretation and exclude more people from the country.
On the other hand there wasn't convincing evidence produced by the government that people from the foreign countries in question posed a harm to the United States. Giving the benefit of the doubt here, if any evidence of this is out there it's likely wrapped up in classified documents, and probably difficult to quickly produce on the timescale that this order came out. Hence the reason that they're granting a temporary stay to the order, giving the government a chance to prepare its case, and not doing harm to the states in the meantime. Anyway, here's the whole thing if you want to read.
There are arguments laid out in there for why the States were allowed to bring the suit. Edited, Feb 10th 2017 8:50am by someproteinguy