We have this stereotypical image of two men in very nice coats standing facing each other while discharging pistols, unperturbed in the face of possible injury or death.
But how do you get there? Maybe this is a stupid question, but how do you inculcate a youth with the belief that he must be prepared to kill or be killed over matters of honor? Did dads give their sons THE TALK and tell them “Whatever you do, don’t be like your Uncle Fitzhugh and run away like a coward”?
This is a really interesting question, as it is both so simple, yet also so complex. In the end, we're looking at the development of a specific view of honor that held great sway over elite male society for several centuries, and which saw the pressure of society place a value on maintaining that honor over life itself. I don't have a specific answer where I've dealt specifically with development of honor as a concept - and to be honest that is less something that is focused on by historians than it is anthropologists, Julian Pitt-Rivers, Anton Blok, and Frank Henderson Stewart are good ones to look into - the best works on this are usually in that field - but I have a few I would highlight to trace this development, and which I believe speak to the heart of your question.
I would first point here which discusses the rise of the duel in the Early Modern Period and the development of the honor culture, in Italy, which underpinned it, while this looks a bit more at the challenge to duel and why it was done. I think that covers a lot of ground of what you are asking, but feel free to bring any follow-up questions you might have.