Maps A, B, and D all depict a straightforward lunge into the Low Countries, which would likely have been the German plan had they invaded between October 1939 and January 1940. Why did the German high command opt for such a straightforward approach? There are several interpretations. Hitler wanted an invasion almost immediately after the fall of Poland, leaving the Wehrmacht little time to prepare a plan. It may have also been a conscious effort on the part of German commanders to present such an unappealing plan that Hitler called off the invasion. Most German leaders were not confident of their ability to defeat the Allies in 1939-1940. Chances against a well-entrenched French army and its allies seemed slim. In Strange Victory, Ernest R. May highlights the strong opposition to the invasion among German officers; a premature or botched offensive in the West might have triggered a military coup.

Map C depicts the alternative plan concocted by Eric von Manstein. Manstein and other officers of Army Group A floated the idea to Hitler, who had not been impressed with original invasion plans anyway. The Mechelen incident provided further incentive for the Germans to adopt a new plan since the original had been compromised.

Now, why was it so successful? First, it wasn't just "a few Panzer divisions," it was seven, plus a motorized corps, followed by infantry. The French had a large army in 1940, though it's reserve divisions were especially poorly trained and equipped. (To be fair, a good chunk of the German army was not much better off). France had 3 light mechanized divisions (DLM) that were roughly equivalents to Panzer divisions. It also had 3 cuirassier divisions (DC) equipped with heavier tanks, though fewer of them. DC also had fewer attached artillery and infantry units, and were generally less useful than a DLM or Panzer division. The 2e and 3e DLM were combined into a cavalry corps, nearly equal in strength to a Panzer Korps. The best French tank, the S35 had a good combination of speed, firepower, and armor and equipped the DLM. In the DC, the Char B, a heavy tank, had great armor protection and good firepower, though low speed and awful range. All French tanks were terrible in terms of crew ergonomics and suffered from having 1-man turrets that forced tank commanders to load, aim, and fire the main gun as well as command the tank. Most French tanks also lacked radios. While on paper a French DLC and a German Panzer division were about equivalent, in reality the Germans had quite an advantage.

In terms of doctrine, the French expected to fight a "methodical battle." This meant a battle in which the supreme commander exercised tight control over reserves to stymie enemy thrusts and open the way for a decisive counter attack. Tactically, the French emphasized heavy artillery firepower to overwhelm the enemy with complex barrages. French infantry divisions had 60 guns organic to the unit to effect that objective. Infantry tank battalions would support the troops on the ground, DCs providing extra muscle for tough fronts. DCR were to exploit breakthroughs or screen infantry until they had a chance to dig in. While this might seem "backwards" compared to blitzkrieg (a term many historians contest) it's not too far off from British and even American doctrine later in the war.

All of these tactical differences are less significant than the strategic imbalance at the start of the campaign. The allies, led by General Maurice Gamelin, adopted a flawed arrangement of troops. Gamelin sent the British Expeditionary Force with 9 motorized divisions into central Belgium to man the Dyle Line in front of Brussels. The French First Army, with 3 motorized divisions, 3 well-trained colonial divisions, and the Cavalry Corps moved into the Gembloux Gap to the right of the British. Finally, he sent the Seventh Army, including the other DLM and 2 motorized divisons, north to Breda in the Netherlands with the hope of creating a unified front with the Dutch army. He also kept over 30 divisions behind the Maginot Line, despite that fact that the line was intended to conserve forces for the main effort in Belgium. All told the allies had close to 60 divisions deployed north of Namur. Opposite them was German Army Group B, with only 29 divisions, including 3 Panzers. Along the Maginot Line the French had over 30 divisions against only 19 low-quality divisions of German Army Group C. But, in the center along the Meuse, the French had fewer than 20 divisions, most of them second-rate, against 7 Panzer divisions and over 50 infantry divisions following behind.

Why did France collapse? The Germans concocted a good strategic plan born of desperation, one that gambled by sending nearly all of Germany's best units through a difficult piece of terrain and then on to a deep penetration of the enemy's rear. They happened to attack at the weakest part of the allied line. The fact that the French had sent their strategic reserve on an errand to Holland certainly made things easier. Aside from that though, French command and control were poor, units frequently arrived on crucial battlefields too late and too disorganized to do much good. Coordinating a four-nation alliance proved impossible. What air power the allies did have they used poorly.

Aside from the aforementioned book by Ernest May, I highly recommend The Blitzkreig Legend by Karl Heinz Freisser whigh breaks down how little faith the German High Command had in the offensive and how the Germans were able to transform a tactical breakthrough into an operational and then strategic victory so quickly. For the French, see Robert Doughty's Seeds of Disaster and Jeffery Gunsburg's Divided and Conquered. Julian Jackson's book is a good overview of the campaign and explains the long-term political ramifications for France. Alistair Horne's To Lose a Battle is often cited but is really pretty dated at this point.