Why are there two abrupt changes in the genome sequencing price curve?
This graph from the Broad's Opinionome blog (ugh) is somewhat more annotated:



As noted elsewhere, the precipitous drop in 2007 is almost certainly due to maturing next-gen sequencing (NGS), in particular Illumina.  Illumina acquired Solexa in 2007, which offered gigabase-level sequencing ability.  In 2014, President of Illumina Francis de Souza essentially said as much:


  During the EmTech conference, De Souza said Illumina’s success was due to a “hard pivot” the company made in 2006, when it got into the DNA sequencing business by acquiring Solexa, a U.K. startup, and bet its fortunes “on a technology with no sales, that no one knew if it would work.”


Essentially, by 2007/2008 you had stellar technology+company hard-selling it = adoption

I'm less certain about the less-dramatic drop in 2014/2015, but I agree with Franck that it's likely largely be due to the introduction of the Illumina HiSeq X Ten in early 2014.  You can read some corporate spam, but it was indeed a big deal.  There were other technologies that probably played a role (new PacBio system released in 2013, etc.) but Illumina was and is top-dog.

None of this takes into consideration increased interest in such research, or the faster computers, larger SSDs, and expansion/normalization of multi-core systems that makes working with such data easier.