I googled the quote and all I could find is a reference to a Christian scholar named Martin Laird. So, perhaps he said it first. I don't know and it doesn't really matter.
I imagine the mountain with all the weather that can hit it: storms that blow with lightening and thunder, rain, snow and then suddenly bright blue sunny skies. It can all happen so fast or then at times it can be dark for days. But there the mountain sits. And yes it may be affected a bit by the weather, a rock or two may fall, but essentially it remains the same. It is what it is despite all that is thrown at it.
I have sat on my cushion meditating and chanting, I am the mountain, not the weather. It feels good. So much of spirituality is just remembering. It helps to meditate just to remember what is true.
Just as the weather changes so do our feelings. But we are not those feelings. They are something that goes on around us, they come and go. And when we are immersed in them it is immensely helpful to remember that they are dynamic. The sun will shine again. But, of course, the opposite is true too.
We are the mountain. Which is not to say we are permanent or even solid. I think the importance of the metaphor is that we are not the weather.