At that moment, Great Britain had been involved in the Second World War since September 1939, so the country was going through a hard time, economically and socially speaking.
Actually, what caused Chamberlain to resign was his recent political attitude towards Nazi Germany.
Contrary to Churchill, he didn't really took a firm stance against Hitler's expansionism: let us take the Munich Conference (1938), for example. Here Chamberlain and the French PM agreed to make concessions to the Nazis, because they believed a compromise would have saved Europe from another conflict. That was the "appeasement" policy. As a result, Hitler was allowed to annex some parts of Czechoslovakia on the condition that he would not occupy other countries.
We now know that Chamberlain's appeasement policies failed and that the Nazis soon betrayed the agreement, attacking Poland in 1939. Moreover, by the time Chamberlain resigned in 1940, the Germans had occupied Norway, making the British Army retreat.
This military defeat and the failure of the appeasement policy had a negative effect on his political reputation. The Allied Powers were then in a difficult situation, while Germany continued to expand on the continent. Finally, Chamberlain decided to resign, lacking of political support.
He was succeded by Winston Churchill, who had already gained enough trust thanks to his strong hostility against the Nazis. He, then, led Great Britain to the final victory.
