The death of a "dear friend" is the question in this quotation from Emerson. Who is the dear friend, and why has he died?
Emerson writes, "To a man laboring under calamity . . ."
In this, he is referring to someone whose mind is weighed down by real or imagined misfortune, adversity, disaster, or trouble.
Next, he writes, "the heat of his own fire hath sadness in it . . ."
This means that the normal energy a person has, which includes the range of emotions from joy to fear, is dominated by sadness. The heat is the product of the fire (normal energy), but that heat is now full of sadness.
"Then, there is a kind of contempt of the landscape felt by him . . . "
So, the person feels sad but also contemptuous of "the landscape," or his surroundings, society, and environment. He resents others, often resenting their happiness—the world becomes sad, too.
". . . who has just lost by death a dear friend."
The dear friend is the god-given joy of living. He's lost his greatest companion, the ability to experience happiness and life.
To paraphrase: "To a person beset by disaster, her own life is like a light that shines on only shadows and despair. So the whole world is darkened, like it is for someone who has just witnessed the death of her best friend."
It's hard to paraphrase because any paraphrase removes the poetry of it.