The simplest answer is that France wound up with a zone of occupation because French political opinion was almost uniformly in favor of this, and a combination of military necessity and political negotiations gave them a zone.

The political spectrum of the Fourth Republic in 1944 was overwhelmingly in favor of some sort of French participation in the upcoming occupation of Germany, although the reasons varied widely. On the more conservative end of the political spectrum, one could sum up the goals as "retribution" - the need to punish Germany for its constant invasions of France, and to neutralize any possible future threats. This went beyond denazification, seeing little difference between the German state and its people, and called for a "deprussianization" - the breakup of Germany into smaller states, and a shift of German political power to the southwest, closer to French control and influence. Edmond Vermeil was a political scientist on this end of the spectrum, and Charles de Gaulle and General Marie-Pierre Koenig were sympathetic to this outlook, although I don't want to imply that they signed up to some sort of comprehensive program.

The French Left was, in summary, more interested in rehabilitation than retribution. French Socialist and once-and-future Prime Minsiter Leon Blum was a leading force in this outlook, emphasizing German disarmament, punishment of war criminals, and social reform, ideally in some sort of Europe-wide federal structure.

It's worth pointing out that the occupation zone that France wound up with was the smallest of the four, and at least initially the most rural. As mentioned elsewhere, the French were not part of such conferences as the Potsdam Conference, but this should not imply that France was completely frozen out of diplomacy among the World War II superpowers. De Gaulle had in fact flown to Moscow in December 1944 to confer with Stalin on postwar Germany and future Franco-Soviet relations (you can read a translated transcript of their conversation here), and this resulted in a Franco-Soviet Treaty of Alliance and Mutual Aid a couple weeks later.

The French zone itself was constructed out of areas originally allocated to British and American occupation (the Rhineland-Pfalz area came from Britain, the southern Baden Wuttemberg area came from the US), and was done previous to the Allied invasion of Germany proper starting in late 1944/early 1945. The French First Army not only made good on occupying assigned areas, but also occupying Stuttgart, which had been assigned to the Americans. The turnover of the city to American authorities was negotiated in return for the transfer of Mainz and Koblenz to French occupation. As far as Berlin goes, the French hoped to have control of the cultural center of the city transferred to them (remember the whole city was occupied by the Soviets at first), but ultimately they had to settle for a small sector in the northern part of the city. The French military government was of substantial size - something like 330,000 in the southwestern zone and 400,000 in Berlin in autumn 1945, although admittedly these tended to be relatively untrained and static forces.

So in summary - there was no participation buy-in level determining who got to participate in the occupation of Germany. France had a strong political consensus in favor of a French occupation, and successful negotiations with the bigger Allied powers secured zones for French forces to move into, which was relatively easy given France's location.

Sources:

J. Reinisch. "The Forgotten Zone: Public Health Work in the French Occupation Zone". The Perils of Peace: The Public Health Crisis in Occupied Germany available here

Karen Adler. "Selling France to the French: The French Zone of Occupation in Western Germany, 1945—c.1955". Contemporary European History, Vol. 21, No. 4 (November 2012)