They intervened because the Dragonborn asked it, and then the only thing they really did was host the peace summit as they were the most valuable location. Every other location would seem to favor one side or the other, and even then they felt awkard doing it. The Greybeards would never involve themselves in a war, the whole reason they live like that is to avoid war. At most they would kick enemies down the mountains, but that's about it.
True, but I do believe that the Thalmor wont just sit by and let the Greybeards exist at the top of the mountain. If the Thalmor were to move into Skyrim and wage a war against them, killing everyone they come across that is not an elf then eventually the Greybeards would be all thats left and I'm almost positive that the Thalmor wont just let them exist in peace. They will attack them and eventually they will defeat the Greybeards, even if they have to starve them out. Because of this I think the Greybeards would join the fight in defending Skyrim, even if its simply to defend their mountain. I dont think the Greybeards would leave Skyrim, at most they would help drive the Thalmor out before retreating back to their mountain, they would not push on and take the attack to the Thalmor's homeland. They wont be happy with it but I do think that the Greybeards are wise enough to see that if they do nothing then they will be defeated. They need to choose a side.