Why was the battle of Bouvines that incredible?

SHORT ANSWER

The Battle of Bouvines (1214) saw Philip II of France defeat a coalition of European forces (H.R.E., Flanders, Boulogne, England) brought together by King John of England. The consensus among modern historians is that Bouvines was one of the most important battles of the Middle Ages. It had far-reaching consequences, including Magna Carta, but there is little evidence to support the contention that it was won by the French King Philip II against "outstanding odds".

However, the achievements of Philip Augustus over the entirety of his reign were considerable. He established the power of a previously weak French monarchy, eroded and then effectively crushed the Angevin Empire at Bouvines, as well as humbling the Holy Roman Emperor.

DID PHILIP WIN AT BOUVINES AGAINST 'OUTSTANDING ODDS'?

According to most modern historians, no. In a review of prominent French historian Georges Duby's Le Dimanche de Bouvines (English edition title: The Legend of Bouvines), Carol P. Jamison of Indiana State University notes that Duby made extensive use of five contemporary texts. Duby

describes the differences between French accounts of Bouvines, which often exaggerate the prowess and victory of the French king, and German and English accounts, which attempt to minimize defeat.

Some nineteenth century French historians claimed that Philip's army was heavily outnumbered. For example, A. Magin's Histoire de France (1860) claims the invasion force numbered 100,000. More recently, this 2014 article on the site of the Union Populaire Républicaine (a Eurosceptic party), Les origines et les conditions de la bataille de Bouvines claims

Le roi de France et ses troupes étaient dans une infériorité numérique marquée.

Translation: The King of France and his troops had a marked numerical inferiority.

but then somewhat contradicts itself by saying that French forces numbered 20,000 as against 24,000 for their opponents. Another article, Bouvines, ou la naissance d'une nation, cites no sources but seems to have been digging around in the aforementioned dubious 19th century French histories as it claims total allied forces numbered 80,000 against 25,000 French (though it is unclear if the text is referring the battle itself). The article adds

Les Français sortent finalement, et contre toute attente, vainqueurs de cette bataille

Translation: In the end, the French, against all expectations, come out winners of this battle

A fair enough observation if the numbers cited are correct, but modern historians do not accept such numbers. J. F. Verbruggen, cited in Wikipedia, gives 6,500 to 7,660 French lined up against 8,800 to 9,000 allied. French historian Philippe Contamine seems to concur with the ratio at least:

En face, ses adversaires n'avaient pas une supériorité numérique évidente

Translation: [Philip's] opponents did not have obvious numerical superiority

John W. Baldwin, in The Government of Philip Augustus, concurs:

It is likely...that the two armies were evenly matched in numbers.

WAS PHILIP'S VICTORY DECISIVE? WAS PHILIP A GREAT STRATEGIST?

Bouvines ended up as a decisive victory for the French, but it was close fought at first and Philip II could easily have lost - he was unhorsed and almost killed at one point. Dividing his army before Bouvines was risky but ultimately paid off. Both contemporary accounts and modern historians speak more of mistakes made by the allied leaders in the lead up to the battle and of the actions of individual commanders during the battle itself than of any great tactics or strategy.

All the same, Philip did effectively counter John's strategy of attacking from two different directions by splitting his forces, sending his son Louis to successfully confront John at Roche-au-Moine to the south. John was forced to retreat, thwarted when key allies led by Aimery VII de Thouars refused to fight. Although Philip's strategy turned out to be spectacularly successful, John Guillermin, in The Angevin Empire, argues that it was forced upon him.

Looking at Philip's reign as a whole, he had a patchy record militarily, generally doing well against weak or poorly-lead opposition (e.g. King John) but often on the run when faced by more able opponent, most notably Richard I. Nonetheless, Philip was adept at exploiting the personal rivalries and weaknesses of others to enhance his own position (and thus that of the French monarchy) and rarely let slip an opportunity when presented with one.

HOW IMPORTANT WAS BOUVINES?

In short, very. On the importance of Philip's victory at Bouvines, French historian Philippe Contamine observed:

the battle of Bouvines had both important consequences and a great impact

Bouvines put an end to the Angevin empire built up by John's father Henry II and, in doing so, enabled Philip to lay the foundations of a powerful French monarchy. Philip's success at Bouvines

safeguarded the administrative reforms he had undertaken since 1190, boosted royal ideology, extended royal control in northwestern and central parts of France at the expense of Anglo-Angevin claims, cleared the path for the Albigensian crusade in the south, and eliminated Flemish and German threats from the north and east; the battle’s ramifications for internal English, Flemish, and German affairs were far-reaching. The battle thus secured Capetian supremacy in France throughout the thirteenth century...

Unseating a Holy Roman Emperor was no mean feat. Otto IV, King John's nephew, lost his crown to Frederick II after fleeing the battle. Baldwin calls this Philip's "most spectacular achievement at Bouvines". In 1218, Frederick proved to be valuable ally of Philip's in the War of the Succession of Champagne. Meanwhile, another coalition commander, Ferdinand, Count of Flanders, languished in a French prison for the rest of Philip's reign.

For England, John France, professor emeritus in medieval history at Swansea University, observes that

Bouvines is the most important battle in English history that no-one has ever heard of...

Without Bouvines there is no Magna Carta, and all the British and American law that stems from that. It's a muddy field, the armies are small, but everything depends on the struggle. It's one of the climactic moments of European history.

While Bouvines led to Philip asserting his authority over the barons of France, it had the opposite effect for the loser, King John, in England. J.C. Holt, a Magna Carta specialist, states:

the road from Bouvines to Runnymede was direct, short and unavoidable

Cited in The New Cambridge Medieval History vol 5

Other source:

The Battle of Bouvines (1214) (chroniclers' accounts)