Virgin Galactic's flights are sub-orbital and pass below the Kármán line (about 100 km up), so technically the passengers don't qualify as astronauts in space, but while they experience weightlessness, this is a consequence of the trajectory of their spacecraft rather than them being unaffected by Earths gravity.
The force of gravity on the occupants of their craft is actually very similar to that acting on us on the surface, but the astronauts are in freefall along with the spacecraft - this is exactly the same as what happens to occupants of the aircraft that follow a freefall trajectory (vomit comet) who also experience what is referred to as weightlessness.
There is little difference between these two situations, and that of someone in deep space (say on route to Mars). However, the article you read is correct and the effect is really an effective matching of the force of gravity with the force that is attempting to keep them travelling in a straight line (as per Newton's first law of motion). Gravity is what is preventing the occupants and spacecraft from following the straight line they would otherwise have.
Theoretically you can experience the same effect travelling on a train at hypersonic speeds on the surface of Earth. If the train was able to travel fast enough you would experience the same thing on a train (even if it was at same altitude throughout). Actually, if it was possible for the train to travel fast enough you could even experience -1g and be sat on the roof of the train (the -1g would in reality be centripetal force being twice the force of gravity, but in the opposite direction, and producing an overall force of -1g).
p.s. Updated answer after obtaining confirmation that they only reached 53 miles altitude. It would seem that the astronaut wings awarded to the passengers are more symbolic than official recognition of entering space (it would seem a different standard it being met here).