It's been a while since I read it, so feel free to seek other replies, but iirc from Atkinson's 'the day of battle' much of the purpose was to press the initial gains made by pushing the Germans and axis forces out of Africa. Italy was seen as a softer road to the underbelly of Germany. Defended by lesser calibre troops. Attacking here posed a few objectives -

A quick drive north could knock Italy out of the war and take the fight directly to Germany's border.

An attack on Italy would help tie up German soldiers from the eastern front and potentially lead to a lesser defended France.

Amphibious assaults and airborne operations in Sicily and Campania helped proved these tactics and offered a tremendous learning experience for the full invasion of France.

Again, as it was more lightly defended the allies were able to attack with the forces in place in 1943 as opposed to hitting more hardened beach heads which required another year of troop build up. With the army already staged and blooded in Africa, Italy was the logical next step.