If a person has gangrene in his toe, why is it necessary to eliminate half of his leg?

Not always. Even if there is only visible toe gangrene sometimes the surgeon is concerned about poor healing of the toe amputation wound. Perhaps the circulation is so poor that an amputation at the level of the knee is more likely to heal. Sometimes the circulation can be improved to increase the chances of healing.
PVD. It depends upon the extent of the PVD in the leg; it's extent and if it can be corrected via stenting or a bypass, to name two.
More information. Generally speaking when an amputation is performed we try to keep as much viable tissue as possible. However, infection (where, how extensive) and vascular status can be a significant driver in the decisions. So while only a toe may be gangrenous, infection may be on bone or caused significant soft tissue loss or there may be inadequate blood supply to the lower limb.
It isn't. The level of an amputation depends on the level of adequate circulation to heal the wound. If a toe is gangrenous the rest of the leg must be investigated to identify the best area of blood flow to do the amputation.