Actually it is clean fog that is created, just plain condensed water mixed with CO2 gas.
The simplest way to make fog is to pour hot water onto a controlled quantity of dry ice in a bowl or bucket. The effect will be immediate but not long-lasting as the water soon cools, which stops the fog creation. The trick is to keep adding hot water, not dry ice, to continue the effect.
A fog machine contains a heating element which ensures that the process is continuous until all the dry ice is used up. Generally 5 kg can fog an area of 2000 sq ft up to a foot deep for up to 10 minutes whereas you would need a lot of hot water refills to achieve this with the ice in a bucket, so for a big event it is always worth hiring a fog machine.
Live music events will often work with more than one fog machine to ensure quick coverage of a large area. The fog can also be piped from one machine to more than one outlet, but check the natural draughts on stage in advance as the fog can easily be diverted and may not go where you expect it to.
If you do hire a machine, beware as a there are “Fog machines” that create just a smoke effect by chemical means which can make the atmosphere unpleasant. Also the smoke will not sit on the floor like dry ice fog does and create that really special effect of walking or dancing on a cloud.