Why is the novel, American Sniper (Chris Kyle) relevant today and what can you gain from reading it?

American Sniper offers an inside perspective on U.S. Special Forces, the nature of war, and the problem of mental health. By reading it, a person can gain knowledge about topics that would otherwise be closed to them, from a point-of-view that they may not have considered before.
Chris Kyle's experiences in the military and as a Navy SEAL, in particular, are the center of the novel. He explains how he became the kind of person who doesn't attach a feeling to killing enemy combatants. Rather, he aims, shoots, kills, and then gets ready to aim again. The lack of feeling for the people he shoots is of particular interest to a reader who likely has not experienced that type of situation. Moving through Kyle's life and experiences in the war, it becomes clearer why he's able to be so emotionally removed. When at first he hesitates to shoot, eventually that hesitation disappears. The reader follows him on that journey and his reasons for it.
The book also sheds light on the nature of war in the modern era. One thing that's discussed by Kyle's wife is that he's forced to fight people with rules that don't govern them. It makes the reader wonder what the better option is—because surely US soldiers shouldn't fight like terrorists. At the same time, though, endangering soldiers isn't a good option either. There are no good answers to the questions of war. Kyle makes it clear that from his perspective, might is what wins wars. The might of the United States (the soldiers who fight for the country) are what allowed them to win engagements and take back cities.
The mental health and relationships of soldiers are other things that make the book relevant. The relationships that Kyle has with his family, friends, wife, and children are all affected by his job. His wife notes that she believed he loved being a SEAL more than her. Kyle had a lot of problems adjusting when he came home from the war because he was haunted by some of his experiences there. This sheds light on the plight of veterans who may not have the resources or support they need to heal from their experiences.