It also adds a lot of tension to the group. In the beginning, they're harassing him for his nationality, and the way he tries to keep himself clean. Then, when he comes up with the plan, everyone, even Towns, ends up having to begrudgingly defer to his leadership.
I love to love my Princess.

Two more thoughts about this:

1. Dorfmann's nationality provides a nice way to explain his career track - particularly, his choice of employer and aircraft designed.
2. Given that the movie was made in 1965, it would be hard for an American audience not to think of Werner von Braun while watching it.

Whatever the reason, it worked perfectly. I assumed in fact that it was written for a German. This was post WWII, and there was not a lot of love lost between the Allies and the Germans. Germans were also known for their engineering prowess, and it was no great stretch to think this "engineer" could redesign the wreckage into a flying machine. Captain Towns was a proud man, and it must have eaten him alive to not only have to give up some control, but to do so to a slightly obnoxious German. Wonderful characters..

In the book the injured person was the German. I think TooSlim is right, it's just that they wanted Hardy Kruger, who did a great job.
For who would bear the whips and scorns of Hollywood (;-p)

As they were using an international cast for the film, I guess they may have done so because they wanted Hardy Kruger. And the idea of a German added the element of dramatic tension due to the dislike of Nazis by the British soldiers (ie, comments about what were you doing in the war, and super race, etc)