In short Germany, and Italy as well, wanted access to the much of the Middle East's vast oil reserves.

A big factor of the war, as with any war, is resources. Access to the vast reserves would allow the war machine to continue.

Most of the Middle East was under control of the Allies, which allowed the British, the Free French and eventually the USA and USSR access to vast amounts of oil.

Germany's plan for the North Africa campaign was to take push into the Middle East and take control of the vast oil reserves, as well as eventually put pressure on the USSR in a second front, with their belly virtually exposed, especially the Caucasus region with well...its oil reserves.

Another big factor is the geography. Control of the Suez Canal, and Egypt in general, would allow control of a very important geopolitical area for freight and shipping.

Why would the Allies want to control North Africa? The same reasons as above, plus it was their territory prior the outbreak of the war, why just hand it over to the enemy?

For Britain specifically, the Suez Canal was the vital lifeline to India and the South East Asian colonies, and the Dominions of Australia and New Zealand.

It was a crucial supply route and India was especially vulnerable to Axis interference thanks to its flowering nationalist movements. They were all vital for the supply of manpower too, especially after the loss of a sizeable chunk of Britain's professional army at Dunkirk.

Britain's focus on a North African war in defiance of US and Soviet urging, it's Pearl Harbour-like destruction of the French fleet in Algeria and the invasion of Italy's not-so-"soft underbelly" was arguably all to this end.