Knights in the battlefield, monks in the chapel. Something more perfectly calculated to fascinate the medieval mind could hardly be conceived. They received privileges, money, land; they found recruitment easy. Its wealth, one of the major elements in the legend that attracts the kooks, became a thing of legend. And other military orders, the most famous of which was the Order of the Kights of St. John of Jeruselm, more popularly known as the Knights Hospitaller, founded by Blessed Gerard Thom, who ran and guarded hospitals and hospices for pilgrims (and tended to have an Augustinian rather than a Cistercian tone).
The Templars came to a terrible end -- another reason they attract crazy people. On October 13, 1307, King Philip of France had all the Templars arrested and interviewed, or, rather, tortured, in order to collect confessions about Templar rites of initiation. Since these rites were conducted secretly, there was no open evidence against any accusation that the enemies of the order might make up; and since confessions were collected by torture, evidence for any made-up accusation could easily be fabricated. Pope Clement V vigorously protested, and declared the proceedings null and void; but with the confessions in hand, Philip was able to convince the Pope to conduct a separate ecclesiastical inquiry, which, of course, investigated the order not just in France, but throughout the world. The inquiry found that the knights of the order, for the most part, were innocent of any charges, and that there was no evidence that the order itself embraced any heresy. The question of whether the Templars should continue as an order sanctioned by the Church was brought before the General Council of Vienne in 1311; the great majority of the conciliar fathers did not wish to condemn the order, but because of the scandal, they felt that something had to be done. To avoid actually condemning the order, it was deemed best simply to dissolve, not as punishment, but as a matter of prudence. According to Clement:
The majority of the cardinals and of those elected by the council, a proportion of more than four-fifths, have thought it better, more expedient and advantageous for God's honour and for the preservation of the christian faith, also for the aid of the holy Land and many other valid reasons, to suppress the order by way of ordinance and provision of the apostolic see, assigning the property to the use for which it was intended. Provision is also to be made for the members of the order who are still alive. This way has been found preferable to that of safeguarding the right of defence with the consequent postponement of judgment on the order. We observe also that in other cases the Roman church has suppressed other important orders for reasons of far less gravity than those mentioned above, with no fault on the part of the brethren. Therefore, with a sad heart, not by definitive sentence, but by apostolic provision or ordinance, we suppress, with the approval of the sacred council, the order of Templars, and its rule, habit and name, by an inviolable and perpetual decree, and we entirely forbid that anyone from now on enter the order, or receive or wear its habit, or presume to behave as a Templar. If anyone acts otherwise, he incurs automatic excommunication.
One of the things I've always been puzzled about is why the Hospitallers aren't as big a part of the conspiracy story as the Templars. After all, the Hospitallers were major rivals of the Templars; on the dissolution of the Templar order, they received Templar property; and, unlike the Templars, they still exist. But that's perhaps the whole point; the Order of Malta does excellent charity work, but it doesn't look like the sort of organization that is running the world behind the scenes. Templar nonexistence functions like the version of Templar secrecy. Templar secrecy allowed the order's enemies to make up anything they pleased and pin it on the order; Templar nonexistence allows the kooks to do the same.
UPDATE: After some discussion, Pensans has convinced me that the claim that the inquiry found "that the knights of the order, for the most part, were innocent of any charges" is probably too strong and certainly not sufficiently nuanced.