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in general, the effective region protected by a lightning conductor depends on its sharpness.
when a cloud carries charges, opposite charges are induced on the surface of the items on the ground. an electric field is set up brtween the clouds and the items on the ground
items which have greater sharpness indicates higher degree of curvature (esp. a lightning conductor) where charges acculmulate in a easier way resulting in a higher charge density. this implies a greater electric field is set up between the clouds and a lightning conductor. when the strength of the E-field is high enough, i.e. breakdown E-field strength (3*10^6 V/m) is reached, air molecules nearby the sharp edge will be preferentially ionized to initiate a chain reaction to further ionized all other molecules along the way from the lightning conductor to the cloud and a large discharge current is resulted.
if a lightning conductor is not installed, the high buildings will become the sharpest items to reach the breakdown E-field strength value at the fastest pace and eventually discharge which cause damage on the building or electric shock to whom touches the building.