There's two elements being confused a bit here, but both are equally common.
The picture above shows someone cleaning off used tires. As mentioned above, this is very common in nearly any racing series that uses race tires. If you go at all offline (and even still if you don't) you will pick up plenty of what is called "OPR", or "other people's rubber". In series that use street tires this is less of an issue because the tires just aren't grippy enough to pick it up.
In amateur racing circles this is normally handled with a torch, putty knife and a beer at the end of the day.
The other item mentioned here is tire shaving. This is used only when street tires are required, and is done using a specialized machine, not normally by hand. The advantage of shaving street tires depends largely on the tire you start with, but generally it is to avoid overheating the tire (tread blocks flex under load, adding too much heat) or to improve the 'feel' of the tire (addressing tread 'squirm' mentioned above). In either case, the tires in these series normally need 3/32nds of tread depth BEFORE the race to be considered legal. Much less than this doesn't really add performance anyways, and just shortens the life of the tire.
Other interesting notes:
1) A lot of series will have one spec tire that is used either shaved in the dry, or at full-depth in the wet (the Toyo RA1 is a prime example - it looks a lot like F1 grooved tires when shaved)
2) Some racers will also hand-cut a rain pattern into a racing slick to use as an intermediate tire
3) Some racers will also hand-cut sipes (basically very narrow grooves) into their tires to cause them to warm up faster. The small cuts allow the tires to flex more internally, adding to the heat generated. You'll see these tires used on either cold days or for qualifying laps.