What is Winston Churchill's argument in the speech "We Shall Fight on the Beaches"?

To understand the argument Winston Churchill made in his famous speech to the House of Commons, we need to understand the context of the surrounding days.
At this early point in the war, England was one of the few countries refusing to appease Nazi Germany's conquest of Europe. Many in the House of Commons believed that Churchill should not have been made prime minister on May 10, 1940, and disagreed with him on key issues. One of these key issues was Churchill's refusal to surrender to Germany's advancing military. On the same day Churchill became prime minister, the battle of France began and quickly turned tragic for the British and Allied armies. On the beach in Dunkirk, France, the Allied forces were surrounded on all sides by the German army. This was Churchill's first test as prime minister, and he believed that it would result in catastrophic loss of life for the British Army. What ended up happening was far different. The British were able to rescue approximately 338,000 men from Dunkirk, despite their initial belief that they would only be able to rescue some thirty to forty thousand men. This victory was a massive one for the Allied forces, as it held off the Nazi war machine for just long enough for the Allies to regroup. It is important to remember that this was before the United States entered the war. With the information that hundreds of thousands of more men were rescued than Churchill initially believed, he gave his famous speech to the House of Commons on June 4, 1940.
In the speech, Churchill essentially talked of the victory at Dunkirk and the consequences of this victory on the immediate future of England. The Battle of Britain was a little over a month away; through his speech, Churchill prepared the House of Commons, the people of England, and the rest of the world for what was to come. Churchill spoke of what the people of England had faced in centuries before as a way to show that they had been tested in similar ways in the past:

Turning once again, and this time more generally, to the question of invasion, I would observe that there has never been a period in all these long centuries of which we boast when an absolute guarantee against invasion, still less against serious raids, could have been given to our people. In the days of Napoleon the same wind which would have carried his transports across the Channel might have driven away the blockading fleet. There was always the chance, and it is that chance which has excited and befooled the imaginations of many Continental tyrants.

Churchill reminded the people of England that the threat of invasion always existed from tyrants similar to Hitler—yet England had always fought and remained. He argued that England would carry on and survive. With this reminder, he also reminded the entire world of what England had endured and overcome.
Churchill began to lead the fight against Nazi Germany's invading armies; thus, he called to arms all those who wished to resist Hitler's tyranny. At the end of the speech, Churchill made clear reference to England's most powerful ally, which had not yet entered the war:

We shall go on to the end, we shall fight in France, we shall fight on the seas and oceans, we shall fight with growing confidence and growing strength in the air, we shall defend our Island, whatever the cost may be, we shall fight on the beaches, we shall fight on the landing grounds, we shall fight in the fields and in the streets, we shall fight in the hills; we shall never surrender, and even if, which I do not for a moment believe, this Island or a large part of it were subjugated and starving, then our Empire beyond the seas, armed and guarded by the British Fleet, would carry on the struggle, until, in God’s good time, the New World, with all its power and might, steps forth to the rescue and the liberation of the old.

This speech is most famous for the initial lines in this conclusion; however, the last lines are the ones that made the greatest call for help. The "New World" is a clear call to the original colonies, now the United States. The "old" is a reference to the "Old World" that is England. Churchill believed that, with this victory at Dunkirk and his people's willingness to fight, the United States would come to England's and the world's aid—and they finally did on December 11, 1941.