Almost all the images shown on that website are done with Digital SLR camera's with far superior lenses. Like the (no longer produced) Nikkor 6mm and 8mm SLR Fisheye. Some are done with "Film" camera's and scanned digital by professional photographers with decades of experience.
These older SLR fisheyes lenses start at around $3000.00 USD secondhand. Coastal Optics will custom grind lenses brand new but you will be looking at around $40,000 to get you started.
Don't expect results this clear from a Nikon Coolpix of any modal (it won't happen!)