Because it's more informative. "a person" doesn't tell anything descriptive at all; "friendly" or "a good father" would be relevant if we were talking about their personalities, but if we're talking about their economic needs, then "homeless" is the single property that's most relevant to the discussion.

With your examples about mental handicap, the same principle applies - it matters what's the topic of the discussion, what's the next sentence, and thus where the emphasis should be. If the fact that that person is mentally handicapped is irrelevant to the rest of the story, then "person" is the part that should be emphasised; but if the mental handicap is the key factor for what happens next, then that is the most salient information of the sentence and it should be emphasised.