Why would foreigners risk their life at war

Do you have any specific war or specific foreigners in question?

I can tell you a bit about (what I think were) the reasons of Pulaski and Kosciuszko.

For them, an idea of freedom from oppression was very important, as their own country, Poland, was under a foreign occupation at the time. For whatever reason they could not fight for their own country, but by fighting for American freedom, they could at least fight for the idea of freedom.

Other reasons could be struggling for fame and eternal glory.

The answer is an appellate court.