1. The trier of fact may be a jury -- or a judge. If it's a judge, it's called a bench trial. There are all sorts of reasons why a defendant may choose one or the other... You might choose a bench trial if the defense is going to be based around a complex legal issue, or a jury if it's a play on sympathy, for example.
2. The judge, even in a jury trial, may have some influence on a finding of guilt or innocence -- but is supposed to be impartial. Any significant influence is grounds for a mistrial, and most judges really don't like to find themselves questioned in an appeal.
3. The anecdote may be true, but I think it's been largely exaggerated for effect. Or the guy was an idiot, and represented himself, and left the door open to be hammered. A halfway competent defense attorney could tear down that "month or a little more" of training to be meaningless. Something like this:
"You've trained in the martial arts?"
Yes.
"For how long had you trained at the time of this incident?"
Around 5 weeks.
"How often did you attend classes, and how long were they?"
Twice a week, for an hour and a half.
"So, that's a total of around 15 hours of training?"
Yes.
"And what has that training consisted of?"
Stances, a few kicks...
.... and so on...
Then call the instructor to the stand... and again play up how little has been learned in that time frame.
So... if the guy really got hammered "just because he was a martial artist" -- there's almost certainly a lot more to the story.