It does snow on Mars. But not where the RSLs are - that's at the poles. You also get mists / clouds of water vapour in the very deepest basins there. But rain as on Earth is impossible, because the droplets would be at close to boiling point, because even in the deepest points, then boiling point of water is no more than 10 C. Think of rain with the rain and air at around 90C - a bit like that - it would tend to dry up before it hits the ground.
But in the past, as recently as 500,000 years ago, Mars had enough water for the flowing water to form gullies (despite what they said in the presentation - it was a bit of a simplification perhaps - the RSLs themselves don't form gullies, and though gullies do form in present day Mars, they form are at temperatures that suggest dry ice is involved)
So - that's because of changing orbital eccentricity and axial tilt - at times it has thicker atmosphere. Right now the parameters are such that even if you liberated a lot of CO2, unless you could get it up to ten times its current atmospheric pressure, it would all fall out as dry ice at the poles. But in the past it was stable with a thicker atmosphere that could permit rain.
So one idea is that the source of the water was replenished some time back and has just stayed as ice ever since - but is hard to get that to work especially for the RSLs that start near the top of peaks..
It could also be that it comes from deep underground, the water. If you had geohermal heating and some deep source of liquid water, that could work.
Or it could be that it forms by deliquescence, where salts, left in moist air, become damp. At night Mars reaches 100% humidity even though the air is so very thin, so this is possible. It also forms frosts in the morning from the high night time humidity - which quickly disappear as the surface warms up. That's in the equatorial regions - Vikings 1 and 2 photographed the frosts.
Though again it is hard to get this to work producing the amounts of water that would seem to be needed for the RSLs.
All three of those main hypotheses have both advantages and issues and I don't think it is settled yet.