I'm surprised that no other answer yet has mentioned that what we have here is a mention, not a use, of the word "dead." Properly punctuated, the sentence would be:
What do you mean, "dead"?
or to use a slightly older punctuation style,
What do you mean — "dead"?
That is, the speaker doesn't know what the addressee meant by the word "dead," and is asking for clarification.
What do you mean by "dead"?
Omitting the quotation marks is just a colloquial shorthand (like using a short pause — indicated by comma or em-dash — instead of the word "by").
The other answerers are absolutely correct that most likely the speaker is trying to convey that he doesn't believe his ears.
He's dead, Jim.
What do you mean, "dead"? You mean, like, unconscious? In a coma? You can't possibly mean that he's actually dead!
Regarding the use-mention distinction, consider the difference between
VADER: Luke, I am your father.
LUKE [incredulous]: What do you mean, "father"?
and
VADER: Luke, I am your father.
LUKE [immediately deferential]: What do you mean, father?