The halogens, particularly in their diatomic free states and within various oxoacids, are strong oxidizing agents by virtue of their high electronegativities, electron affinities, and reduction potentials. The polarizability of the heavier halogens also makes them almost uniquely versatile as both good leaving groups and strong nucleophiles, depending on conditions. Additionally, the relatively low bond dissociation energies of the halogens contribute to their aptitude to react by free-radical mechanisms (coupled with the exothermicity that usually accompanies the propagation of those reactions).
In their diatomic forms, they readily add across the $\pi$-bonds of alkenes and alkynes (which are ubiquitous and essential structural features of myriad types of biomolecules), and they can oxidize numerous other functional groups under the right conditions and in combination with other substances. Various oxoacids that incorporate halogens are often extremely strong oxidizers as well (household bleach, for example, is an aqueous solution of sodium hypochlorite, which is mild by comparison to certain other halogen-containing oxoacids). In those reactions, they typically serve to add oxygen to molecules in place of, e.g., hydrogen, while themselves serving as leaving groups (or parts of leaving groups).
The antiseptic properties of halogens (and halogen-containing oxoacids) are directly consequent to those chemical properties. At least one source suggests that elemental iodine, for example, reacts by mechanisms similar to the ones described above to disrupt key structures of bacterial enzymes, proteins, and lipid membranes: it oxidizes the thiol groups of cysteine or methionine to disulfides, interfering with protein synthesis; it reacts with unsaturated lipids to alter their structure; etc. These are biological equivalents of the very same reactions that are routinely carried out in organic chemistry labs. If you want more details/examples, they should be easy to find with a web search.
In short, the very same awesome chemical reactivity that makes them so indispensable in organic synthesis, also means that they are highly reactive towards (and hence, usually harmful to) all manner of organisms.