At first it was not an invasion, the British were trying to stop the Germans from reaching the Suez canal which allowing them to have access to vast oil fields. Once the Americans joined the war, they invaded North Africa so they could eventually invade Italy which was considered the soft underbelly of europe.

Prior to the American involvement in the war and after France's defeat, the British were on the defensive and they focused their efforts in North Africa to protect the Suez canal which was a large source of their oil imports from Iran which were obviously vital to the war effort. Once the U.S. got involved, the idea was to deal with the German forces in Africa which were a much easier target, because the British Navy was making their resupply difficult, especially when compared to the dug in troops in Europe. Then the idea was to attack German held Europe from Italy which was considered to be the weakest point in the German defenses which proved true when the allied forces faced Italian forces although they faced some difficulty with the German reinforcements once they arrived. On other theory is that the Allies were also trying to delay the opening of a second front in Europe because they knew that the war in Russia was sapping the strength of both sides and they wanted to face a weaker Germany when they did invade France. This has also been argued by some as a part of a more long-term strategy in looking ahead with regards to the USSR and trying to make the war cost them as much as possible so that they would be weaker at the end of the war. The U.S. and Britain were known to be enemies of the USSR (they had sent troops to fight against the Bolsheviks during the Russian Civil War following the Bolshevik Revolution).