That site is not a good one. I threw it up here and should have read it a little more before linking it.
This guy's blog explains the origin and thinking behind dry fasting in more detail and references a few studies which are focused on Ramadan. The idea is similar to regular fasting in that you're creating a more stressful environment in your body to promote culling weaker cells and boost fat loss (where the body supposedly extracts more water from within itself).
https://www.marksdailyapple.com/guide-to-dry-fasting/
If you're not eating during the fast, is it really necessary in the short term to have water to break down nutrients when none are being ingested? From my understanding of the dry fast, it's not supposed to be done under extreme conditions or physical duress. It seems doubtful that you're going to incur significant damage to your joints from a short period of abstinence. Wouldn't the main area of concern be damage to your organs, not joints?
Not arguing for or against this but want to cover all sides of the issue before settling on it being bunk.
Anecdotal loosely related story: when I was 18 I lived with a retired pro cyclist and his wife for 2 weeks who actually laughed at me for taking too much water on a 2 hour training ride claiming that he'd finished a 240km mountain stage in the tour de France with only 1 bottle in a year where he placed 5th overall. Not the same thing as dry fasting but still interesting.