The primary reason in my opinion is that it was the first of the grand tours. As everyone knows the concept of the tour was the brainchild of a French newspaper editors desire to increase the circulation of his publication. Of course there is tremendous cachet in being the pioneer of anything.
The language of cycling is French - derailleur, velodrome, domestique, peloton, cadence, all French terms.
The French have been more important in administrative terms in the sport of cycling and in sport in general. The Union Cycliste Internationale was founded in 1900 in Paris. The first President of FIFA was a Frenchman and it was founded and headquartered in Paris. Jules Rimet was the third President of FIFA, also a Frenchman and the man credited with conceiving the World Cup tournament. The founder of the modern Olympics and the IOC was Pierre de Coubertin, a French aristocrat.
Then there are other political reasons - France is a larger country, with a bigger economy and population than Italy and Spain. Geographically it is nearer to the heart of Europe and it is more important and influential in international affairs - at the time of Napoleon it was the pre-eminent continental European power, it had a very big colonial empire, it is a nuclear power and a permanent member on the UN security council. Lastly, although the Nazis installed a collaborationist regime after the fall of Paris France did not suffer the stigma of the long term association with fascism that Italy and Spain did/do.