Establishing Petain as the new head of France was simply a sensible move. Petain, as the overall commander of French forces even before that, was able to compel the vast majority of the French military to cease military action. Most notably, for instance, is the fact that it effectively removed the French Navy (which was essentially intact at the time of the armistice) from the board. If the Axis powers had continued to prosecute the war, the French Navy would've simply sailed to Gibraltar and effectively joined the Royal Navy for the duration of the war. Instead, the French Navy essentially sat at anchor until Mers-el-Kebir (when the British bombed it). And even after that particular instance, the French Navy basically sat in Toulon until Germany renounced the deal with Vichy France and invaded, at which point the French scuttled the ships rather than allowing Germany to capture them.

In addition to removing the French Navy from the board (at least for two and a half years), it also drastically reduced the odds of France's overseas troops defecting. And, for the most part, it kept large numbers of those men from defecting. It wasn't until the dissolution of the Vichy regime and occupation of North Africa that de Gaulle finally started gaining steam. While de Gaulle would show up in battles very late in the war (long after the matter was decided), France had ceased to become a meaningful participant in the war in 1940 and that wouldn't change until Darlan's defection in 1942 (at which point he took a significant number of Vichy troops in North Africa with him). Darlan, interestingly enough, probably would've wound up heading the Free France Forces if he hadn't been assassinated shortly after his defection.

There were other reasons, ranging from the economic to the political, but the neutralization of the French military almost overnight was definitely the biggest among them.