This is a great question! I believe the most commonly accepted answer for this is that iodine acts to oxidize enzymes involved in the cellular respiration. It blocks the transport of electrons in the electron transport chain, which would result in the cell being unable to make ATP to survive. Iodine may also interact with amino acids in the proteins of the cell membrane. Reactions of iodine with certain amino acids could result in the proteins changing shape or possibly becoming denatured, rendering them unable to perform their specific task.

Iodine in a solution with bacteria is reduced to iodide. Reduction to iodide suggests that an oxidation-reduction reaction has taken place. Reactive groups within proteins on or in the bacterial cell are oxidized in this process. Oxidation of proteins important for bacteria respiration and cell membrane structure prevents the cell from making energy and maintaining its integrity. Thus, it dies! The exact mechanism is however not entirely agreed upon, but we know it works.

Here is a link to an old paper where scientists studied the interaction of iodine and bacteria.