This is a terrible graph, layout-wise (if they're arguing that higher gasoline prices reduce miles driven, why is the dependent variable on the horizontal axis?), but it is interesting. The sections where the graph bends backwards are gas price spikes that lead to reductions in miles driven per capita. When gas prices go up, people do respond by driving less, and the bigger the price increase, the bigger the drop in miles driven.
The most interesting part of the graph is the section from 2008 to 2009, where the price falls while miles per capita continue to fall. I don't know what's causing this. If I had to guess, I would guess that, unlike the 1970s, we have a greater ability to adapt to gas price changes--by telecommuting, maybe--so once people started identifying such opportunities, they just kept on going. Also, the drop in housing prices may have allowed some people to move closer to work. Finally, the 2008-2009 price, while lower than 2007-2008, is still high by historical standards, so people may still be trying hard to find ways to reduce their high gasoline bills.
One more thing: If we want people to drive less, and use less gasoline, gas prices are the tool to use. Higher gasoline taxes will drive the price up and reduce gasoline usage. Whether or not that is a good idea is a separate question, but if the goal is to force people to use less gasoline, higher gasoline taxes are the most efficient tool to accomplish this.