First of all the weather is generally clear. The average annual rainfall at the top of Maui’s 10,000′ Haleakala is about 38″ with most of it falling November through April. That provides for an acceptable number of rain days in each month but the rainfall for the Big Island’s summit of Mauna Kea is even better, way better. The total annual rainfall is only about 7.5″. On both mountains a typical trip to the top from sea level will have you ascending through a ring of clouds floating like a halo in mid-trip and eventually rising above the clouds allowing you to see them below from the top as though they were a soft white billowy blanket. On these days the view to the heavens is unobstructed.
The second reason has to do with the atmosphere. It is very difficult to see through even the largest and most powerful of telescopes if the air is too thick in the areas surrounding it. The higher you are, the thinner the air so it makes sense that the 10,000 and 13,000′ heights would fit the formula of what would work best. But that is not the only condition required of the atmosphere enabling it to be a good viewing spot. The air can’t be too damp and the temperature differential between the air at that height and the air strata below it must be gradual or it will produce adverse conditions that will distort viewing. Place a check by Hawaii’s tallest mountains regarding the perfect atmosphere for an observatory.
Light pollution from nearby cities is not helpful. Fortunately the islands of Maui and the Big Island are not populous and there are no surrounding land masses producing light pollution for thousands of miles.
Although not on the equator, the Hawaiian island are fairly close and as such they have a good view of both the northern and the southern skies. If an observatory is located to far north or too far south it will only see that half of the sky. Hawaii has the big picture.