The main reason is that strong acids and bases undergo complete dissociation in water while weak acids/bases undergo partial ionization
Remember that in order for a substance to conduct electricity, there should be a presence of freely-moving charged particles. In the case of acids and bases, charged particles are produced when it undergoes dissociation to form their corresponding ions.
In acids, it refer to hydronium ions and the corresponding conjugate base while the ions produced by dissociation of a base are hydroxide ions and the corresponding base.
However, since strong acids and bases undergo complete dissociation, more ions will be produced resulting to a greater conductivity compared to a solution containing a weak acid/base.