6.1-30.5mm is the actual focal length (zoom) range for the lens, but the sensor in the camera is much smaller than that in a dSLR or film camera, so the focal lengths are also smaller. The G15 uses a 1/1.7" format sensor, so its crop factor is roughly 4.5x. The spec to look for here, if you know how focal lengths translate to field of view (FOV) on film is the "35mm equivalency". For the G15, this is 28-140mm. That is, the field of view on the G15's lens is similar to a 28-140 lens on a 6D, or an 18-88 on a 700D.
1:1.8-2.8 is not the aperture range of the camera's lens, it's the maximum aperture of the lens. Because the lens is a zoom lens, two numbers are given: the first one is for the wide end of the lens, the second for the telephoto end. So at 6.1mm, you can open up the lens to f/1.8, but at 30.5mm, you can only open it up to f/2.8. You can, of course, always use a smaller aperture setting than the maximum.
The 5x Zoom factor is simply calculated by dividing the longest focal length by the shortest to get a relative measurement of how large the zoom range is. In this case, 30.5/6.1 is a 5x zoom. However, note that both a 1-50mm and a 100-500mm lens would also be 5x zooms but vastly different in nature, which is why the absolute description of focal lengths rather than the relative measurement of zoom factor is typically used for purposes other than marketing.