In the second para, Arnold compares him to dwelling in "Heavens of Heavens" akin to God, and when other mortals try to understand him, they can see only the base of his "lofty knowledge" cloud. He is like a mountain whose top most part has been covered by clouds, and only what is visible is the base of cloud.
Shakespeare wrote so much about nature, but ironically he did not have schooling experience rather nature was his teacher. Here Arnold says he was a self taught man and had so much of confidence in himself that he did not depend on any other person to get this confidence. External honours and praise from others did not matter, as he was a self secured person.
The last paragraph is so beautifully written, what it means is Shakespeare is the sole voice for the people who suffer and who are weak. He is the voice of the people who are unable to express their pain and emotions, indeed Shakespeare became their voice that would help alleviate the pain of the common immortal spirits