I have a basic question: Once the data output is generated from a self-taught computer, do the researchers know HOW the computer reached its conclusions as to what regions of DNA are biologically relevant?
Unfortunately I think the answer to your question is simply "No".
This is because that link says;
"The system uses an artificial intelligence technique called deep learning ..."
And what is meant by "deep learning" in AI terminology is AI learning using "deep neural networks" which are neural networks with multiple hidden layers. One down side of using neural networks as opposed to using, say, an AI knowledge based system, is that it is generally very difficult to know HOW it reached its conclusion because that information is highly implicit and, especially if its an extremely complex neural network like deep neural networks are, tends to be 'hidden' in the neural network.
In general, if you want to know HOW the computer reached its conclusions you must avoid using neural networks altogether but then avoiding neural networks altogether generally makes it a lot harder to get your computer to recognize complex patterns in noisy and fuzzy data.
One of the things I am currently researching is a way around that using a new kind of AI 'logic' (too complicated to explain here) that should allow an AI, that is represented as software only in a conventional computer so it is NOT neural network, to STILL, despite NOT being a neural network, efficiently recognize complex patterns in noisy and fuzzy data AND then to still tell you clearly HOW it reached its conclusions. If you know anything about AI you should know that is a highly difficult non-trivial task and you may well be very sceptical I could achieve this.