DDT can change into many different forms that are only slightly different from the original. We call these new forms metabolites. We think that one of these metabolites, abbreviated DDE, interferes with certain reproductive enzymes in birds.
Enzymes act as helpers for certain chemical reaction. They can speed up biological processes, slow them down, etc.. In birds, certain reproductive enzymes affect how much calcium is deposited in eggshells. DDE probably gets in the way of some of these enzymes. Consequently, eggshells have less calcium, which makes them easier to break.