This question is a case study for anachronism, or rather how knowing what happened next influences our interpretation of historical actors at the time they make their decisions. In this case, we are looking back from Blitzkrieg France 1940 rather than ahead to a repeat of 1914.
Hitler decided in early 1938 that it was time to start the war for Lebensraum. As evidenced by the way the war started in 1939, it didn't matter too much what Chamberlain did. The war started when Hitler decided it was time. At the end of September 1938, Hitler got cold feet about starting his war. Chamberlain allowed him a way out.
When Chamberlain returned to London waving the piece of paper in his hand, it was not the Munich Agreement. It was Hitler's signature that Germany would "behave as a big boy" and have its grievances addressed through diplomatic means, i.e. not through aggressive war. Hitler did not intend to sign it, but was outfoxed by Chamberlain in the reception that followed the signing of the Munich agreement.
So why did he do the Munich agreement and why the piece of paper?
Look back at what happened 1914 to 1918, and then to 1939 to 1945. If you're going to go through that, you want to be damn sure that the cause is just. No doubts. No shadows.
In 1938, Hitler's cause celebre was national self-determination, the liberation of oppressed German minorities whose desire was to come "Heim ins Reich." The Anschluss with Austria. The liberation of the Sudeten Germans. The Daniziger Germans. Causes that seem morally right, that Germans understand to be just causes.
If this doesn't make sense in 2014, then step back and look at Putin in Crimea. An "oppressed" Russian majority that "wants to return to Russia." Nobody could dispute that the Crimea was Russian until Krushchev arbitrarily handed it to Ukraine SSR in 1954. Russia has nuclear weapons. Is it worth starting a potentially world destructive war over that? What if one could point at injustices to Russians in the Donbass? If Putin were to make a stink and then invade in order to "rescue" them, would it be worth fighting a war?
Hitler's propaganda was that he was liberating oppressed ethnic Germans. Chamberlain offered him to put his money where his mouth was. The rationale was to make Hitler prove beyond a shadow of a doubt that his true intentions were aggressive.
BTW the UK was in no condition to fight a war in 1938, and knew it would not be ready for several more years. They had high hopes for their navy and "sanction" power of the blockade.
Also, Hitler and Chamberlain both knew it would be a couple years before the UK and France were otherwise ready to do much militarily. It was more about easing Germans psychologically into the war mode, keeping supply lines open as long as possible, and keeping options open as possible as to when exactly to attack Western Europe before turning eastward again.
In the end, Chamberlain got to look the chump, but in return go to war with Britons grimly behind him.