The objectives were more political than strategic. Fundamentally, it was “make sure our Alliance doesn’t suffer an embarrassing military defeat”. The vast majority of Axis forces in North Africa were Italian, not German, despite the publicity given the latter, and officially, the whole shebang was under Italian command with the Germans being a supporting power. (Unofficially, the whole unified chain of command thing didn’t necessarily happen)

The problem was that the Italians, hoping to expand its African holdings, took advantage of the apparent British weakness by deciding that Egypt seemed an awfully appealing place to set up shop, and surely the British, especially after the defeat in France, would not be in a position to stop them.

They failed to take in account a few minor details, though. One was the fact that London was not prepared to cease its support for the Western Desert Force, and though there was some controversy in the UK over the decision to send materiel to Egypt instead of keep it in the UK after the horrendous losses in France, it seemed to do the WDF no end of good. Secondly, 7th Armored Division, the core of what would become Eighth Army, had been well trained by arguably one of the best practitioners of armored warfare in the British Army at the time, Percy Hobart. (Back then it was known as “Mobile Force(Egypt)”). Hobart’s story is fascinating and too few people know his name, but that’s a story for another time. Thirdly, the Italian Forces themselves weren’t particularly vested in the whole operation, neither were they as well equipped.

The upshot of this all was that the Italians started getting their arses kicked, and after some six months of Italian/British fighting it became obvious that in order to prevent an embarrassing failure of its ally, Germany would again have to stick its oar in and lend a hand. (See the Balkans campaign a little earlier).

Although the defeat of the British in North Africa would have perhaps opened up opportunity to head for British Palestine, then into Iraq and the oil fields there, and maybe from then the Caucasus in a second front, that wasn’t the initial intent. It was simply to stop the Italian collapse.