Was 14,000+ year old DNA "laying around in cave dirt" protected from degradation, or is it just naturally this robust?
To be clear they are not finding long DNA strands but short fragments. DNA becomes more robust the shorter it gets. Fragments can last millions of years. The local chemical conditions have a huge influence on how long DNA lasts the oldest being hundreds of millions of years old under ideal conditions. 

The ones the archeologists are finding are not even close to enough to rebuild an intact strand but it can be used for matching. Of course contamination is a big problem, anything that dies leaves DNA in the soil. With humans it becomes especially problematic becasue the archaeologists themselves are a major source of contamination.

For an indepth look at contamination protocols, some of which as still questionable, can be found here. Sorry its behind a pay wall. 

Likely they are referring to deamination cytosine substitution, which creates mismatched pairs of DNA and occurs more and more as DNA ages.