The obvious answer is because FDR picked him. Well, why did FDR pick him in December 1943? First, Ike was Supreme Commander of the Allied Forces in the MTO. Secondly, FDR had decided that Marshall had to stay in DC and be Chief of Staff. Furthermore, the British, especially Churchill knew and liked Ike. So he got the job.
Since his being chosen to lead the North African invasion in July 1942 lead more or less directly to his being chosen to be the Supreme Allied Commander in the ETO. The question is how did Ike go from being an obscure COS of the 3rd army in December 1941 to being in charge of the invasion of North Africa in July 1942?
The answer is luck, ability and politics. Ike was brought to the Pentagon in December 1941, because his friends of 20 years, General Gerow and Mark Clark wanted him. Once there he was able to impress General Marshall with his ability. After his 2 1/2 years as Chief of Staff MacArthur's aide, Ike knew what a COS wanted in a subordinate. When Ike went to England in May & June of 1942 he got along extremely well with the British, especially Churchill. Not only was Ike a born politician but he had years of experience in DC, the Philippines, and Paris in dealing with foreigners and politicians.