Here we compare the photo on the left to an atmospheric sounding, taken from Wilmington, OH (about 7 miles from the storm) right at the time this photo was taken.  The red line shows the temperature profile of the atmosphere, and you can see that it steadily decreases with height up to about 13 km, after which it begins to warm again.  This portion of the atmosphere is known as the tropopause (marked by the blue arrow).  A parcel of air lifted from the surface will also cool, but not as quickly as the atmosphere.  It's temperature will follow the brown curve in the diagram.  As you can see, the surface air parcel remains warmer than the surrounding atmosphere until it reaches the tropopause.  Since warmer air is more buoyant, it will continue to rise (think of a hot air balloon).  This stream of rising air is the updraft of the thunderstorm.  Usually, an updraft will quit rising when its temperature becomes warmer than the environment (at the tropopause).  The tropopause thus acts as a lid, and the rising air will fan out in all directions and form an anvil.  If the updraft is strong enough, it can punch past the anvil into the warmer environmental air and form an overshooting top as seen here.