The Phoney War (Sitzkrieg, Drôle de Guerre, etc.) seems destined to remain one of the great mysteries of history. It is difficult to comprehend now, after the fact, how such an astonishing combination of missed opportunities, wishful thinking, and indecisiveness on the part of not just one, but two great powers, could have carried on for more than half a year.
The seventh episode of the 1998 documentary series Sworn to Secrecy: Secrets of War is devoted to Sitzkrieg: The Phoney War, and a good introduction. The period is also the subject of numerous books and papers— not to mention various conspiracy theories, and certain narratives of Western betrayal, especially in Poland. Full coverage is not possible in the space of an answer here, but this excerpt from William Shirer's The Rise and Fall of the Third Reich covers most the popular theories:
[D]efeatism [among] the French… the memories of how France had been bled white in the First World War… the realization by mid-September that the Polish armies were so badly defeated that the Germans would soon be able to move superior forces to the west… the fear of German superiority in arms and in the air. Indeed, the French government had insisted from the start that the British Air Force should not bomb targets in Germany for fear of reprisal on French factories.
Fundamentally the answer to the question of why France did not attack Germany in September was probably best stated by Churchill. "This battle," he wrote," had been lost some years before." … The price of those sorry Allied failures to act had now to be paid, though it seems to have beeen thought in Paris and London that payment might somehow be evaded by inaction.
I will provide a bit more detail on three factors:
The British and French governments held Hitler to be a bully, willing to instigate border skirmishes and bark rhetoric, but not start a full-scale war over Poland. In fact, Lord Halifax, the British Foreign Minister, believed that Hitler was about to back down; on August 31, hours before the outbreak of war, he said he had seen in Hitler “the first view of the beaten fox.” So much for that.
Both Britain and France had been re-arming in anticipation of future conflict, and the French had begun to mobilize their army as early as August 26, but the process was incomplete. French commanders reported that they would not have sufficient resources to mount an offensive until 1941–42, and even if that was an exaggeration, other papers have argued that early in the war, the British and French believed time was on their side because it would give them time to coordinate and mobilize the full strengths of their overseas empires. The calculation was not to defend Poland in the short term, but to defeat Germany in the long term.
The British forces were inadequate for mounting a full-scale offensive. The air force was concerned about bombing, because it lacked the means to stop retaliatory raids; the navy could not operate freely in the Baltic Sea; the British Expeditionary Force was quite small compared to the French army. And even the last would take took several weeks to cross the Channel, by which time Poland was already doomed.
Still, Germany had deployed most of its forces in the east, and at the Nuremburg trials, their generals testified that had France and Britain taken action early in September, the course of the war would have changed and Germany might well have been defeated.
2. Misunderstanding of modern warfare
According to William Shirer's The Rise and Fall of the Third Reich, France was bound under treaty with Poland to attack Germany within three days of the order to mobilize, and to launch a major offensive within fifteen.
Gen. Gamelin was not a timid general, but he believed that any attack required an artillery barrage, and made his offensive and defensive plans according. The French army waited in the field while fixed artillery could be brought out from storage, shipped to the front, and assembled, and wanted for planes and tanks. A direct attack on Germany in the north was not possible without violating the neutrality of Belgium and the Netherlands; in the south, the French army did invade the Saarland on September 7 to fulfill France's treaty obligation, but did not advance far into Germany, stopping short of the Siegfried Line fortifications. And shortly thereafter, the Supreme War Council decided not to proceed with invasion, and ordered the army to retreat back behind the Maginot Line.
3. Fear of wider war and hope for a peaceful settlement
And why would the Supreme War Council do that? The horror of World War I was very much in the minds of European leaders. Fear and wishful thinking led them to hold out for what now seems like a foolish amount of time.
At the outbreak of war, Germany was allied with the Soviet Union, something that became clear when Soviet forces joined the invasion two weeks later. Chamberlain and Daladier did not want to risk angering Stalin and widening the war. They might have been able to send forces to reinforce Poland from the Mediterranean, but were not yet at war with Italy, and did not want to risk provoking Mussolini.
At the same time, Hitler was intimating with diplomats that Poland would at last appease him. On September 19 he declared in a speech that he had no war aims against Britain or France, and on the 28th Germany and the Soviet Union issued a statement that the matter of Poland having been "settled" (through their conquest and partition), there was no further cause for war. And the British were in contact with disaffected Germany military officers, hoping they would influence or overthrow Hitler. It did not come to pass.