Cold, dry, arctic air was already in place, but temperatures were still in the mid-upper 30s when the moisture arrived.
As the moisture moved in, when the first flakes fell from the cloud and into the dry air, they evaporated. But as water evaporates, it takes heat energy from the air to do so, thus making the air colder.
As the process repeated, the air became colder and the temperature dropped low enough to make it snow near the surface. Closer to Seattle, the air wasn't as dry, so the temperature didn't drop as far when the rain moved in, keeping it as rain.
You can keep an eye on this yourself. If you see low dew points (the temperature at which the air saturates) around 27 or under, it can be a sign it could snow, even if the temperature to start is over freezing, as the temperature will tend to cool toward the dew point.
Precipitation intensity can also affect snowfall. Sometimes heavier showers can bring colder air from higher altitudes toward the surface, making it cold enough to snow at the surface. This frequently happens in Puget Sound Convergence Zones, where heavy showers can drop the temperature from 38 to 31 in a jiffy.