In short, degree days are just a truncated temperature variable that only counts temperatures above 29C (84F) for each day of the growing season. All temperatures below 29C (84F) count as zero, while temperatures above 84F get counted as the difference to 29C (e.g., a temperature of 30C gives 1 degree day, a temperature of 31C gives 2, etc). Being 10C above the threshold for one day is equivalent to being 1C above the threshold for ten days.
Both graphs give the weighted average of the weather in all counties of the contiguous United States, where the weight is average corn production (i.e., if county A produces twice as much as county B, the weather of county A gets weighted twice as much as well). This weather measure is tilted towards places that grow a lot of corn (Corn Belt) and not representative of the entire US.
The year 2012 has received a lot of media attention for how hot it was (shown in blue below, the second hottest on record to date). Historic baselines for 1950-2011 are shown in grey. Note that 2013 in red is even hotter than 2012, although most of the extreme heat came early (June) or late (August and September), but July was relatively moderate.
Historically, hot years are also very dry. The blue line for 2012 in the next graph is the second lowest. Not so for 2013, again shown in red. While it was exceptionally hot, it was also very wet!
Once the yield figures are out, this should allow us to test whether more water mitigates the damaging effects of extreme heat. By the same token, moderate heat, which is good for crop growth, was way above normal as well is 2013.