It's BS really. They are either lying or just experiencing placebo.
Performance is going to come from proper macronutrient balance and micronutrient intake. Most of those vegans don't get enough protein. That's terrible for cycling and recovery. Most don't get enough fat. Fat is an essential nutrient for cell rebuilding as well, proper organ function, and testosterone production.
Carbs are the only nutrient that's not essential. But they are the body's preferred source of fuel. So great for cycling obviously. They are also muscle sparing so again, great for riding.
If you can eat vegan and get enough fat and protein to hit your macro goals then that's fine and awesome. But it's harder to do that being vegan. Most vegan protein is weak as well. Not a balanced and complete amino acid profile.
So I have nothing against veganism. I just have a problem with thinking it has magical properties that defy the laws of thermodynamics and nutrition.