Shelley thinks that the moon is weary because of its color, first of all.
But the reason he thinks it might be weary is because only the moon, he says, is faithful and constant. He says that all the other things up in the heavens are so different from the moon that the moon can find no companionship.
Because of this, the moon is tired of constantly looking for something that it cannot find.
This is why, to him, the moon seems like an old woman on the point of dying -- "she" is tired of wandering pointlessly around.