When a golfer gets a far worse result than their handicap, what should
happen to their handicap?
His handicap should increase.
The handicap is calculated by taking a golfer's score for the past ten rounds, calculate an average score, and depending on what par is, subtract the average score by par and there's your handicap. The term "scratch golfer" is given to those who average par or better over ten rounds.
For example, here are the last ten example rounds for an example golfer:
92-92-92-92-92-92-92-92-92-92 = 920. 920/10 = 92.
If par is 72, then 92-72 = 20. This golfer would have a 20 handicap.
Now, assume this golfer's next round was a 112 (ie, "When a golfer gets a far worse result than their handicap"), then:
92-92-92-92-92-92-92-92-92-112 = 940. 940/10 = 94.
If par is 72, then 94-72 = 22. This golfer would now have a 22 handicap.
Does it get lower?
As shown above, no.
What does the disqualification mean?
I believe your interpretation of the statement in context is misconstrued.
For example, if you shot a 90 but write down that you shot an 89, then you would be disqualified (given you're playing in a sanctioned tournament) for submitting a scorecard with a lower score than actually shot. If you write down that you shot a 91, the 91 would stand as signed.
Rule 6.6d of the USGA Rules of Golf and the R&A Rules of Golf addresses this (verbiage from the USGA Rules of Golf):
d. Wrong Score for Hole
The competitor is responsible for the correctness of the score
recorded for each hole on his score card. If he returns a score for
any hole lower than actually taken, he is disqualified. If he returns
a score for any hole higher than actually taken, the score as returned