How antibody kill an antigen?

They don't usually: Antibody does not usually kill an antigen but binds it. The attachment may then, under certain circumstances , triggers the release of certain mediators or the influx of body cells (cytotoxic lymphocytes, macrophage etc) to do the job of killing. This is a very simplistic answer to a complex question however.

It does not: It binds to the antigen. Neither are alive. The presence of the antibody may fix complement (inviting phagocytic cells and perhaps activating the membrane attack complex which punches holes in things), attract the attention of T-cells able to kill the cell, or render an infectious particle harmless by coating it. There are more mechanisms. Best wishes.