How did France become a republic?

Actually, that depends on when you are talking. The first French "republic" arose from the French Revolution. Although that Revolution is typically described as starting in 1789 (either with the calling of the Estates General or with the storming of the Bastille), the "republic" arose only after Louis XVI attempted escape and capture 1791 and the formal abolition of the monarchy in 1792. Although "formally" declared a republic [to compare it with the non-autocratic American republic], this "republic" could actually be considered a series of dictatorships (the National Convention, the Committee for Public Safety, the Directorate, the Consulate) as compared to the electoral concepts which are now considered to be the heart of a "republican" system. The "first republic" formally terminated when the "first consul," Napoleon, declared himself emperor.After the final defeat of Napoleon at Waterloo, the Bourbon monarchy was restored to the French throne. In 1830 a new revolution occurred, which installed a replacement king (the Orleanist Louis-Phillipe) and instituted a "constitutional monarchy" as opposed to a "republic." This government had a series of elections and governments, but it was marked by substantial political chaos. This constitutional monarchy was replaced during the Revolution of 1848 with the "second republic" which formally terminated the monarchy. This "republic" actually involved as much (or more) political instability as the constitutional monarchy which preceeded it. Ironically, Louis-Napoleon Boneparte was elected the President of this "second republic" in 1848, which he overthrew in a coup in 1851 (becoming emperor, Napoleon III) of the second French Empire.With the defeat of the French in the Franco-Prussian War in the early 1970's (Napoleon III was actually captured by the Germans in 1870) and the failure of the Paris Commune in 1871, the Empire was formally dissolved and the "third French republic" came into being. This republic remained in existence until the French defeat in World War II, and is probably the first governemnt which most people would identify as "republican" in the current context of the word (as opposed to being a government which simply is not a monarchy, which is the older "republican" concept).Following the defeat of Nazi Germany, in an effort to distance itself from the bitter heritage of that defeat, a new French constitution was drafted and the "fourth French republic" came into existence in 1947. This republic, which incorporated a very strong legislative control of the government like the earlier "third republic" faced very difficult political and economic problems, which culminated in a an army revolt in Algeria. The army threatened to parachute people into "metropolitan" France unless DeGaulle was named President. DeGaulle would agree to that step only upon the issuance of a new constitution, which substantially increased the power of the French President. To distinguish this new constitutional scheme, this new constitutional format is said to have given rise to the "Fifth French Republic."