The mechanism behind why more fiber helps is pretty straightforward:

Insoluble fiber speeds up gut motility. Faster gut motility means less time for toxins to sit and absorb in your gut.

Also, fermentable fibers serve as substrate for gut microbes, producing short-chain fatty acids (butyrate is one - a primary fuel source for colonocytes - the cells that line your colon).

It also lowers colonic pH, inhibiting pathogenic bacteria.

Lastly, (although there are tons more benefits I'm not listing), soluble fiber is incredible for people trying to lose weight, as highly fibrous foods increase satiety, keeping you fuller for longer.