TLDR: Appeasement was a reasonable policy reacting to reasonable German demands.

I make this argument to my students when teaching the run up to the Second World War.

Ignoring the internal politics of Germany (i.e. whether we should have done anything to prevent the worsening conditions for Jews in the 1930s), the reasons that British policy of appeasement was a reasonable policy are two fold:

First, the British public was exhausted from the First World War. It's hard to explain the war weariness of the 20s and 30s, but if you can imagine the war weariness of Americans about 2008/2009 for the Iraq war, and then multiply that by an order of magnitude (or three), that would come close. We're talking four years of war in which battles counted their losses by the hundreds of thousands, not by the individual, with names that are etched into the consciousness of many a Brit even to this day, names like Paschendaele, the Marne, or the Somme. Moreover, Chamberlain was a democratically elected leader, responsible to the war weary populace. When he returned from Munich, declaring to have secured 'peace for our time', he was greeted by cheering crowds. Suffice it to say, appeasement was a popular policy at the time, and as they say, all politics is local.

The second reason that appeasement was a reasonable was because the demands the Germans were making were not, in light of the declared principles of the League of Nations and the Versailles Treaty, all that out of bounds. In order, the Nazis:

1. Re-militarized the Rhineland - why is it such a problem for them to put their own army in their own back yard?
2. Annexed Austria - the people who populated this German speaking rump state left over from the former Dual Monarchy mostly wanted to join Germany anyway. Hitler himself was from there. And it was national self-determination anyway, one of the major principles of the Paris Peace Conference.
3. Invaded the Sudetenland and then Czechoslovakia - For the Sudetenland, again it was an issue of national self-determination, or so the Germans argued. Again, a significant portion of the local populace wanted to join the Reich. The full invasion of the rest of Czechoslovakia is less easily defended. I would call this the nadir of appeasement.
4. Invaded Poland - Hitler says in his Table Talk that he half expected the British to back down from their commitment to Poland as well, but of course, they didn't and the Second World War began here. I should point out that each of these moves made by Hitler and Germany were portrayed as the 'last time'. Hitler constantly said he had only peaceful intentions, and wanted only for Germany to be an equal party with a seat at the table. Given that it was the most populous country west of the USSR and the economic heart of the continent, there was every reason to believe (perhaps hope?) that Hitler was telling the truth.

With hindsight, however, we know that the main assumption that Chamberlain was working off of was incorrect - Hitler wanted a war, and would do whatever necessary to get his war. But given the circumstances at the time, I think any democratically elected leader would have been forced to do basically the same thing.

You might also be interested in this past post, in which a user put the 1930s decisions in context with previous European diplomacy and assumptions while showing why they failed in the end.

Yes, it was a mistake because it was based on false assumptions. Appeasement only works if both sides at the negotiating table genuinely mean what they say and honour their signature to what they have agreed. This was not the case with Hitler.

The appeasers misjudged Hitler. The British Foreign Secretary from 1938-40, Lord Halifax, said in March 1938 that he "distinguished in his own mind between Germany's racial efforts, which no one could question, and a lust for conquest on a Napoleonic scale which he himself did not credit". Neville Chamberlain read a book on Nazi Germany's social and political policies (The House That Hitler Built) that concluded that Germany was preparing for war and that war was the logical outcome of Nazi policy. However Chamberlain wrote after finishing it: "If I accepted the author's conclusions, I should despair, but I don't and won't". Britain's ambassador to Germany, Nevile Henderson, read Mein Kampf where Hitler laid bare his plans for conquest. However he still believed that once Germany had been appeased of its grievances over Versaille, "Hitler and the reasons for his existence and the methods of his regime would disappear".

Therefore, the problem was not lack of information regarding Hitler's intentions; the problem was the self-deception of the British appeasers. In their view to not take Hitler at his word would be "cynicism" and any attempt to construct alliances with other European states to deter German expansion would be a reversion to the "wicked" and "immoral" policy of the balance of power. In fact, the rulers of Britain in the 1930s were singularly ill-suited to diplomacy. I remember one historian saying the rulers of the British Empire between the wars were like tee-teetotallers inheriting a brewery.

Preventing any one state from dominating the Continent had been an English/British policy for centuries. If one state managed to dominate the Continent then Britain would not be able to resist that state by her own. Therefore England/Britain resisted attempts by Philip II's Spain, Bourbon France, Napoleonic France and Imperial Germany to dominate Europe. Facilitating Nazi Germany's expansion, which was what appeasement was, therefore went against traditional British policy and her own national interests.

Re-militarized the Rhineland - why is it such a problem for them to put their own army in their own back yard?

If French and British troops held the Rhineland, the bridges and bridgeheads on the east bank of the Rhine, a German attack (such as in 1914) would be prevented. The Rhineland was an extremely important strategic area. Germany's industrial heartland, the Ruhr, would also lay at the mercy of the Entente if they held onto the Rhineland. France would also have been able to fulfil its obligations to the East European states of the Little Entente.

Annexed Austria...Invaded the Sudetenland and then Czechoslovakia

Austria had never been German territory. Annexing Austria also allowed Hitler to outflank Czechoslovakia. The Sudeten Germans had also never been part of Germany. They had been included in the Czechoslovak state because without the Sudeten territory it would have been defenceless. If Britain and France had gone to war in 1938 over Czechoslovakia instead of in 1939 over Poland, they would have been much better placed. Williamson Murray's book, The Change in the European Balance of Power, 1938-1939. The Path to Ruin (Princeton University Press, 1984) showed how Hitler's invasion of the remnants of Czechoslovakia yielded immense quantities of raw materials, industries, gold reserves and armaments (ready made guns, ammunition, tanks). The Luftwaffe in 1938 could not have taken on the Czechs, French and British all at the same time. The Wehrmacht was not as battle-ready as it would be in 1939. As Murray states, Chamberlain and his fellow appeasers "made the wrong choice on almost every strategic and diplomatic question that they faced".