What properties (if any) do AT rich regions of DNA have?
With regards to DAPI itself, one could tell that the molecule itself has a shape complementary to the shape of A-T rich DNA. So in this case it's "special" because DAPI happens to have the structure to form bonds to A-T rich DNA regions. (There can very well be another organic molecule that binds to C-G rich regions). A quick google search will show how it looks like 



With regards to A-T in general, A-T share 2 hydrogen-bonds. However, this is not the reason why it's easier to break A-T rich regions. A-T rich regions destabilize the DNA double-helix. Refer to this http://nar.oxfordjournals.org/content/34/2/564.long. Transcription initiation occurs near T-A rich regions, e.g. TATA box in eukaryotes and Pribnow box in prokaryotes.