Why do you need to subtract when using a climb/descent performance chart for an altitude other than sea level?
If you look closely at the example, it's describing a situation where you're cruising at 5500 feet, and landing at an airport with a "pressure altitude" (elevation) of 680 feet. The chart, however, is indexed to zero-altitude. You can't just reduce your cruise altitude by the elevation of the airport, because performance varies with altitude (indicated by the curvature of the arcs for fuel, time, and distance). 

To find what you'd need to descend to the airport, you must run the numbers twice - once for descending to sea level from cruise (3 gallons, 9 minutes, 30 NM), and once for descending from the airport to sea level (1 gallon, 2 minutes, 4 NM). Note the two sets of dashed arrows on the chart, providing the two example calculations. You then subtract the numbers for the airport from the numbers for the cruise altitude, as you would hopefully not descend below the altitude of the airport!