Upshifting, it isn't usually necessary, though I'm not a street racer, so I can't comment on the validity of that particular quote [0].

Downshifting, particularly into first as you're coming to a stop sign, it'll save a fair amount of wear on the synchros. Or if you have a car with worn synchros, it'll help ensure a quicker, smoother downshift in general. A 5-3 downshift on an offramp usually benefits, for instance.

[0] Given the substantial liberties they took in general, your skepticism might be justified.

If you double clutch near redline on your RPMs, you can get marginally faster gear changes with less wear on your synchronizers.

Double clutching is changing into neutral, then into the target gear, right? How could that be faster?

Try it (shifting at high RPMs, specifically near redline) in a Honda Civic (or any other decent, but not higher end car) and you’ll see. Shift at normal RPM ranges and you’d be right to feel doubtful it would be faster.