Chamberlain resigned as Britain's prime minister. After making a pact with Hitler trying to keep the peace, giving Czechoslovakia to Germany, Germany kept attacking nations. In Chamberlain's leadership Britain was not able to fight back against the Nazis, so people stopped believing in him.
Winston Churchill was First Lord of the Admiralty, even before he was prime minister he was loud and clear in his belief that Britain needed to be armed and go stronger against the threat, his actions and his conviction at the beginning of the war, plus his skills in military strategy and leadership were more than enough for him to be named the new prime minister.