EDTA has high affinity towards divalent ions like Ca2+, Mn2+, Mg2+ which are cofactors for many active enzymes inside the cells. That includes nucleases which digests DNA molecules. Once the cell is disrupted, nuclear envelope goes off and the nuclear content comes into contact with the cellular content which is rich in nucleases. So the broken cell is treated with EDTA to chelate the ions so that nucleases loose their function and that we are able to get good yield of DNA.