First off it should be noted that this was 100% an Allied operation with 3/5ths of the beaches assigned to British and Canadian units.

But in general the opening of a 2nd front in Western Europe was essentially a prerequisite for the Soviets to remain in the war. The situation in Italy meant that comparatively few German divisions could defend against a much larger Allied force, so another mean of relieving the stress and sucking forces away from the East was needed. The commitment had also been confirmed by FDR and Churchill to Stalin and their last conference in Tehran.