The "vibration reduction", "image stabilization", "optical stabilization" or whatever the manufacturer calls it is a function that uses small optical elements to counter the small erratic movements that the photographer introduces when handholding the camera.
When handholding a camera without this stabilizing function a rule of thumb is to not use a shutter speed slower than 1/[focal length in mm] seconds. Using a slower shutter speed is likely to induce blur from the movement of the camera during the exposure. A stabilized lens counteract this to a certain degree.
Assume that a lens with stabilization turn of gives sharp results hand held at a shutter speed of a 1/60th of a second. Turning it on could very well push that limit to 1/15th of a second. This allows you to gather 4 times as much light that can be used for other things (such as closing down the aperture). This is what people mean when they tell you that the vibration reduction is giving you the extra playroom.
Of course the number of fstops that you gain by turning on the stabilization is not calculated by rather a result of the photographers experience as it depends on the shooting techique and the resolution of the camera among other things.