Octane is a measurement of how resistant a particular blend of gasoline is to pre-ignition. This means that the higher the octane, the less likely it will be to spontaneously ignite. This is what we call "knock". Knock is when the gasoline in the cylinder ignites before the spark plug fires. This results in a shockwave inside the motor that can damage the internals, like the pistons and wreck a motor if left unchecked. KR, or Knock Retard is a measurement of how much the engine's computer is retarding, or dialing back, the timing of the ignition when the knock sensors detect it, to prevent the knock from happening.
So, why do we need higher octane gasses that are more resistant to knock? Harsh conditions inside of the combustion chamber are generally what cause knock. Things like boost and high compression are two situations where motors require high octane gasoline, because boost tends to cause knock as well as high compression. The higher octane gasoline allows the motor to push more air into the motor (boost) or allow the motor to "squeeze" the air/fuel charge more before ignition (high compression) without the gasoline igniting before the spark plug fires. Both of these things result in more power, not the high octane gas.
That's a big misconception that had a role in this video, the high octane gas does NOT give the car more power or performance.. in fact sometimes it can reduce performance. You cant just put high octane gas into a Toyota Camry and expect to have it produce more power. The high octane gas allows the car to run boost with a turbo or supercharger, or run high compression, and THOSE things provide more power.
The video seems to insinuate that the high octane gas is just a sham perpetrated by the gasoline companies to rape our wallets and provide no benefits.
Grand prix's can run lower octane gas if you stay out of boost, and the engine computer can retard ignition timing to help stave off knock. Your car probably does the same thing, so with the boost it's probably better to run higher octanes, but the car can cope with lower.