This is from a British perspective, but needs to be mentioned that many British politicians, notably Churchill, realised the threat that Germany posed with their rearmament. After the horrors of the First World War, the public just wasn't interested in going to war. The German rearmament started in 1936, and at this time the British public had other things on there mind, for example the death of George V.
The people in power realised that to go to war, with a non aggressive county would be an extremely unpopular opinion, and with an election the next year, the passive response can be argued was more for reelection.
At Versailles, the Allies had stated that the Rhineland would remain demilitarized for c. 15 years at least, pending good behaviour (i.e. reliable payments of reparations) on the part of the Germans.
Since getting the Germans to pay their reparations reliably was virtually impossible (in 1924 eventually, under the Dawes Plan, the United States gave the Germans money to pay the French and British, who would in turn pay their debts to the United States), the Lausanne Conference of 1932 had suspended reparations payments indefinitely. As a result, when the Germans remilitarized the Rhineland, 15 years had already passed, and the Anglo-French armies were in no position to resist, and so there was no serious resistance.
When the Germans re-introduced conscription, the western powers had already pretty much disowned Versailles, and had no intention of enforcing it anymore. Besides, would YOU, as president/prime minister of France or PM of Britain, go to war with Germany a decade and a half after the conclusion of the last war, just because the Germans re-introduced conscription, and re-militarized a territory that was unquestionably German, and was going to be re-militarized anyways?
It's worth noting that, for all of its obvious flaws, both with hindsight and at the time, the policies of Appeasement broadly had the support of the populations and public opinion of Britain and France. It was only really in March(?) 1939, when Hitler broke his promise and occupied the rest of the Czech lands, that things began to change.