One of the reasons for using the "zoom during exposure" technique is that it helps eliminate background clutter, but the main reason is that it's fun.
Using a digital SLR makes it easier because you can immediately evaluate your exposure and make any needed corrections instead of waiting for film to be processed. And interchangeable zoom lenses for DSLRs have easy-to-operate zoom rings that compact digital cameras lack. Also, compacts may not let you zoom during exposure.
Pick a subject and make your exposure reading, but select the smallest possible aperture in order to produce a slow shutter speed. You may need a neutral density filter filter to slow the shutter speed even more.
Zoom zoom: This unmanipulated image was captured in the Akihabura section of Tokyo using a Canon EOS Digital Rebel and 18-55mm f/3.5-5.6 kit zoom lens. Exposure was 1/6 sec at f/29 in Aperture Preferred mode and ISO 400. A minus one and one-third stop exposure compensation was applied. © Joe Farace
If you want some things in the image to be sharp, use a tripod. Immediately after you start moving the zoom ring, trip the shutter and keep zooming after the shutter is closed so the overall effect will be smooth. If you zoom in from the longest focal length to the shortest the effect will be different than if you zoom out from the shortest to the longest.