Q1: What does "none the worse for wear" mean?
That he got through the exercise of whatever he was doing without apparently suffering. I don't see from the context here what exercise that could have been, in his cabin, but the phrase might make more sense in another context: "He went out into the city to fight his way through the tangle of traffic and returned, appearing none the less for wear. " That means that he looked just as good or comfortable or rested when he reappeared as he did when he left.
Q2: Why did the journalist stress it was a "man-size" breakfast?
The journalist is emphasizing the subject's size, obesity. Remember earlier he mentioned his "multiple chins"? Apparently he was so fat that his chins (plural emphasized) hung down in flabs of fat onto his chest.
I would have said man-sized, but that's not the question here.
Another person might have eaten "a light breakfast" or a "delicate breakfast" but this chomper filled his plate.