How did the campaigns in North Africa, Sicily, the Atlantic, and in the air help the Allies prepare to defeat Italy and Germany?
It's a bit complicated.....
For the Atlantic, you would know that it is referring to the Battle of the Atlantic, which involved the U-boat campaign against the British cargo vessels which had aimed to launch a blockade on Britain and it was part of Hitler's plan to starve Britain into submission and surrender.
However, the failure of this in the Atlantic paved the way for the Western Allies (USA & the UK) to launch Operation Overlord which was a large-scale invasion on the beaches of Normandy on 6 June 1944. This allowed the Allies to open up a third front against the Axis (I consider the African campaigns as the 2nd front), and hence put more strain on the German Army (Wehrmacht) and its supply lines, having already suffered heavy losses in the East against the USSR.
It took a while to get things together in fight against Germany and Italy.
Africa had a lot of British Colonies in it and the Italians were invested in keeping theirs. There was a fight for Ethiopia in 1940. Vichy France had a French Colony in Syria that was encouraging the Arabs to fight against the British. Britian invaded and neutralized the threat. The Suez Canal was important to the British as a connection to India and the Far East if they could get rid of the Italian Navy.
Many people think of the War as Northern Europe and the Pacific but it was truly a global war. Rommel’s objective was was to help the Italians but capturing the Suez would have been a good thing and getting to Iran would have been a big bonus in the Eastern Front.
From May 1940 to June 1941 Britain was fighting the war pretty much single handedly and had lost most of its modern equipment in France. It was trying to fight the Axis any way it could. It didn’t have the wherewithal to invade France. It also needed to modernize its tactics and build new weapons. They also fought in Greece in 1941 to hold back the Germans who were preparing to invade the USSR.
Clearing out North Africa knocked the Vichy French out of the War as well although they were half heartedly fighting. This also increased the free French troops available. North African Moroccan units fought in Italy. Every success had encrimental gains. Eleven different countries had troops at Monte Casino.
Italy was one battle that paid off the least as it can be seen as a German defensive success. Although Hitler pulled 2 SS panzer divisions to hold Northern Italy, in case the Allies broke through there, out of the Battle of Kursk effectively ending that offensive.
The British African Colonies contributed greatly to the British War effort. There is a number of Nigerian and other African Countries dead buried in Burma. South Africa sent an armoured division to North Africa and Italy.
The Airwar from Germany was an effort that was the equivalent to a 2nd Front that the Russians were begging for. The Russians didn’t see the effects until they got to Germany.
The British work hard with mixed results to get Resistance fighters going in France, Norway and other European countries as well as in Greece and Yugoslavia. British Commando Raids made life exciting for the German defenders.
The Americans reluctantly agreed that the Allies were not ready to invade France until 1944. So then what to do. Thus the Air War and North Africa and Italy.
The Atlantic campaign was simple survival for Britain and the invasion of France needed Britian as a staging ground. The Atlantic campaign was still touch and go until 1943.