Responsiveness is something hard to quantify, but it's a real thing. For an inexperienced rider it might compare to cars, where the differences can be big.

First there's steering response, or how quickly and precisely a car will turn in response to steering inputs, both haw much force on the wheel, and how much you move the wheel. A large family sedan or SUV will be slow to respond and fight to hold it's preexisting straight line course, on the other hand a sports car will be nimble and quick to turn easily. Auto makers and writers test for this with things like a slalom course marked out with traffic cones, and the nimble car can run the course at much higher speed than the less nimble one.

There's also throttle response, brake response, and all other measures for how quickly and precisely the driver's inputs translate to action.

So using the same concept with bikes, you'll find that some are much more responsive in terms of steering and general handling, and more rear triangle stiffness makes them more responsive to your power inputs (but not magically faster, which is a function of engine power).

BTW- while responsiveness is a good thing, it's not a deal where more is always better. More responsiveness may come at the expensive of comfort on rough roads, or long term rider fatigue, and we also have a word for too much responsiveness -- twitchy.

That's the concept, though as I said it's hard to quantify, so there's usually plenty of advertizing fluff in the claims. Read these with a grain of sane (or more) and think in terms of the big picture of what kind of bike they're selling and to whom it's targeted.