Just as you feel acceleration in a car, or the Astronauts felt lifting off of the Moon.
It’s tough to wrap your mind around but the bubble doesn’t actually move. You need to break out of the mindset that there is some sort of definite, unchanging, grid of “space” that you “move” through.
Because you are changing your “at rest” state relative to local gravity. With a warp drive like this, it is that “local gravity” that is moving. You’d feel nothing.
In the future, when we can warp space-time and travel faster than light, will people still be unable to tell the difference between your and you’re, or loose and lose? Or will our telepathy eliminate the need to spell?