Leading up to the start of World War II, France was considered the foremost military power in Europe. They had more tanks, a vast advantage in artillery and were about even in terms of men with Germany. France had also built extensive fortifications along their border with Germany, the Maginot Line. Diplomatically, France had military alliances with Czechoslovakia up until Hitler annexed it in 1938 and with Poland, as well as their tacit alliance with Britain.
When Germany invaded Poland in September of 1939, Britain and France immediately declared war on Germany. While the main body of German forces was occupied with defeating Poland, the French launched a large offensive into German territory known as the Saar offensive. But French military doctrine focused on defense and WWI style trench tactics which most of the world expected to be repeated in the new war. The Allied strategy for the war was to sit behind their fortifications and wait for a British blockade to starve the Germans into submission. As such, when the French offensive encountered the small German screening force left behind, which they could have easily defeated, they retreated back behind the Maginot Line. This gave Germany time to defeat Poland and then begin massing troops along the French border in a period known as the "Phony War".
German Planning for the invasion went through several stages. Most German generals, while more optimistic than their allied counterparts, expected the war with France would be a protracted affair and were certainly not expecting the complete victory they ended up achieving. Initial German plans called for basically a repeat of the WWI Schleiffen Plan with a massive thrust through Belgium to get around the French flank and envelop their army.
However, General Von Manstein, heavily influenced by General Guderian (The "inventor" of Blitzkrieg tactics) drew up a different plan which centered around a massive armored thrust through the Ardennes forest towards Sedan.The Plan was known as "Sickle Stroke".Here's the Allied plan for defense, Sedan is at the bottom of the image. This plan was dismissed outright by German High Command as being too risky. The big problem was that the German armored columns would be completely open to air attack and flanking movements as they moved through the Ardennes which would take days. If the allies did ANYTHING they would be able to inflict heavy casualties on the Germans. But if it succeeded the Germans would punch a massive hole in allied lines and avoid a protracted war. In the end, the plan was approved after Von Manstein personally appealed to Hitler, who by coincidence, had suggested an attack on Sedan earlier but had been rebuffed by German high command. Sickle Stroke was very much an all or nothing plan. A failure would have been disastrous for the Germans, but it offered their best hope to avoid a protracted war.
The Allies for their part, had based their defense around the idea that Germany would simply repeat the Schleiffen plan of World War 1 and planned to deploy large portions of their armies in Belgium to stop the Germans. Unfortunately, Belgium wanted to retain neutrality and thus prevented the allies from deploying in Belgium until German troops entered Belgium, delaying Allied forces considerably.
Obviously, when Sickle Stroke was launched on May 10th 1940 it succeeded fantastically. The French had only placed light cavalry divisions in the Ardennes as they didn't think the Germans would risk moving through the forest. The reported German movements in the Ardennes were dismissed as a diversionary attack and the French sent their reserves to what they thought more likely avenues of attack.
The French Army was at a massive disadvantage vis a vis the German army in terms of communications. The French Commander, Gamelin, still insisted his orders be delivered by motorcycle as he distrusted Radio and many French unit lacked Radios or radio operators. In comparison, the German Army was constantly in contact with its commanders. Every panzer had a radio which aided the Germans greatly in maintaining cohesion in their armored column as they pushed through the Ardennes.
After breaking through the Ardennes, The Germans advanced on Sedan and used pontoon bridges to cross any rivers in their way. The speed of the German advance shocked the French who were unable to react in time. In particular, the armored divisions commanded by Rommel and Guderian were known for outrunning their supply lines and moving faster then German HQ thought safe.
France rushed to contain the situation as German troops started breaking through and moving to encircle their entire army. But French generals moved too slowly. Weygand (Gamelin's replacement as Commander of the French Army) tried to relocate his HQ many times only to find Panzers already waiting for him when he arrived. France's lack of specialized armored corps hurt them badly here. Despite the spirited effort of Charles De Gaulle's armored division which inflicted several German defeats, no French mechanized force had the strength to stop the German panzer corps. German forces moved North from Sedan to the channel and encircled the majority of the allied army while penetrating ever deeper into France. German High Command was shocked at their own success. Some Generals feared the whole thing was an elaborate French trap and they would be hit by a counter attack any day. Hitler himself ordered Guderian's Panzers to halt as they advanced on Dunkirk in order to "consolidate their position" despite Guderian's protests. In the meantime, the British successfully evacuated several hundred thousand troops.
The German successes continued however and with the surrender of most of the French Army after they were encircled either in Belgium or in their positions on the Maginot Line their victory was assured. France sued for peace and was left with no choice but to accept the harsh German terms.
tl;dr: The German Victory was far from inevitable as it is often portrayed in video games and history shows. The German advantages were their widespread communication equipment and advanced combined armed doctrines which utilized planes and armored columns to achieve breakthroughs. But the French Army was well entrenched and the Germans equal or superior in terms of pure firepower. The German plan for invasion was all-or-nothing in nature and it could have gone very differently.
For further reading try "The Rommel Papers" by BH Lidell-Hart, "Panzer Leader" by Heinz Guderian or "To Lose a Battle: France 1940 by Sir Alistair Horne". Also try watching the Vasili Luga episode. All of those shows are very informative.
