They modified an analysis program so that it uses a fixed-size buffer in a vulnerable way (no bounds checks, etc). Then, they synthesized some DNA that, when sequenced and analyzed with this program, overflows that buffer.
To their credit, parts of the paper are very upfront about this, but it's still very hokey, in my opinion. All the DNA stuff feels like a smokescreen around a pretty boring buffer overflow exploit that wasn't even present in the original code.