I understand that many protein DNA binding domains bind to DNA via basic residues such as Arginine and Lysine. But what is the mechanism used to bind to DNA and where on the DNA would these residues bind?
This question has no particular answer. There are several families of DNA binding proteins, some of them bind specifically (e.g. restriction enzymes like EcoRI which binds to and cuts GAATTC or transcription factors like lac repressor which binds a particular point of the e coli genome about 25 base pairs long) and some of them bind semi-specifically (e.g. zinc fingers which bind GC rich regions), others non specifically (e.g. the sliding DNA clamp).
For all of these proteins, they will have a tendency to have positively charged amino acids facing the DNA because the DNA's phosphate backbone is negatively charged. There are lots of protein motifs (like the helix turn helix motif) that fits into the major or minor groove of the DNA. The lysine and argenine are important because they are the positively charged amino acids. Histidine is also positively charged, but it is not as long and flexible, and doesn't seem to fit as often or as well.