Dunno what they mean, maybe nothing, it is sort of a fluff word.

Personally, what I would mean is
- When you increase pedaling force, the bike accelerates quickly
- As you continue pedaling hard, the bike reaches high speeds quickly with relatively little effort from you
- The bike is very sensitive to tiny changes in body lean and steering input, changing directions quickly

The first is about light weight and stiffness, the second is about low rolling resistance (tires mostly) and low aero drag (body position mostly), the third is about bike geometry, light weight, and to some extent skinny tires.

And context matters, a responsive mountain bike would be a slug compared to a responsive road bike.