I've written about the breakthrough at Sedan here.

The common misconception is that the French relied on the Maginot line to stop the Germans, and the Germans simply went around it. In reality, the entire point of the Maginot line was to deter the Germans from going that way, to force the main battle to occur in Belgium and not France. The bulk of the French mobile forces were intended to sweep into Belgium and form a defensive line to stop the German advance. This would prevent the iron mines and industry in northern France from falling to the Germans as they did in World War I.

As to why there was no resistance afterwards, there was quite a bit of resistance. The problem was that the bulk of the French army that might have offered resistance was north of the German breakthrough. This meant that they were cut off only only from supplies, but, because of the French reliance on wired communication, they had poor contact with the French command south of the breakthrough. Crucially, the French sacked Maurice Gamelin, their military commander, on the 17th and replaced him with Maxime Weygand. This paralyzed French operations for at least two days, during which the Germans were able to bring up infantry to shore up the flanks of the breakthrough. By then, it was too late for the French forces in the north to break out to the south. The French counterattack on the 17th at Montcornet and the British counterattack at Arras on the 21st both failed to achieve their objectives.

The Germans paused to reduce the French, British, and Belgian forces north of the breakthrough, allowing the French time to regroup and form a new defensive line. The French continued to try to get to the trapped forces, attacking across the River Somme near Abbeville for several days, but failed due to poor coordination between the armor, infantry and artillery and failure to aggressively exploit initial gains.

The Germans resumed the offensive south on 4-5 June. Despite pulling some troops from the Maginot line and recovery some troops evacuated from Belgium, the French were now faced with defending a long line from Sedan to the Atlantic coast. The Germans had nearly complete air supremacy by then, and were able to attack French forces, particularly the artillery, at will. At this point France simply had too few troops holding too long a line to withstand the concentrated German attacks.

Doughty, Robert A The Breaking Point: Sedan and the Fall of France, 1940

Alexander, Martin After Dunkirk: The French Army's Performance against 'Case Red', 25 May to 25 June 1940