/u/the_direful_spring is mostly correct but I want to expand on it because its a question I have often ask myself.

There were certainty some people who really believed in the strategy of 'taking away' Britain's empire as a strategy to get them to the Table. This idea is very old, Napoleon had the same plan as well. This sort of idea of how to fight war was certainty still common with German officers. While you always have to be careful with the Auto-Biographies of the German Generals, Guderian at least claimed that such was his opinion at the time.

However I think the Hitler and the Nazis never had a real interest in a traditional empire and wanted an Eastern expansion, not colonies in Africa/Asia.

In large part the initial German troupes were sent to make sure the Italians did not collapse. This seems evident by the relatively small numbers that were dispatched and the orders to do basically that. And for the most part, I think that would mostly have been it. Problem was that Rommel not caring about his orders simply took the offensive and totally suprised the British. They had basically moved all the troupes including the Commander of Operation Compass away, mostly to Greece and left North Africa totally open with the assumption that 'no attack could come for a couple of month' (British Commander Wavell explained his actions like that). The failure to finish up North Africa is in my opinion on of the most significant strategic failures of the whole war, but that's another debate.

As it often is with military success, this success allowed him to present that front as a viable way to attack Britain. Rommel now well known and a friend of Hitler constantly demanded more resources and the OKH often reluctantly gave them, but wanted him to stay put, and he off course ignored those orders and pushed the attack only to eventually bog down and demand more stuff. However he never truly had enough to really have strategic level impact.

For me the whole German policy in North Africa really only makes sense if you think of it as compromise between different factions. It was not put in a wider strategic concept of how to win the war. For many Nazis it was a sideshow only useful to occupy the British. For some German Officers it was a better way to victory against Britain then to destroy the Soviet Union.

I also suspect that from a certain point the German Army simply had a 'we don't retreat' mentality even if there were no larger strategic objectives. The Germans putting way to many resources into North Africa that were then lost in one of the most massive surrenders of the Germans in WW2.

A bit of both really. The Western Desert Campaign really began in September 1940 with the Italian invasion of British Egypt. This was initially successful and in October Alfred Jodl, the chief of operations staff for the German military High Command, proposed that the Italians could aim to take the Suez in order to cut Britain off from India, Australia, New Zealand and British East Africa. This was in the light of the fact that by this point it was becoming increasingly clear that the Luftwaffe was loosing the Battle of Britain making the ill-conceived plan to attack Great Britain itself, Operation Sea Lion, clearly nonviable. However the British counter attack of operation compass in December left the Italians in disarray. By going into the new year of 1941 it was pretty clear to the Germans that the Italian forces there were collapsing, retreating back into Libya with the 10th army being encircled and destroyed in February. So it could be said that the collapse of Italian forces in North Africa was what prompted the Germans to deploy the Afrika Korps but taking the Suez was a long term goal from even before they deployed themselves.