Some folks like the military life, even given the dangers, time away from home, and absurd living conditions. Politics has nothing to do with it.
For everyone else, if they do disagree with the politics, then no matter what they are doing, they are still defending their homeland, even given the more peaceful alternatives.
But what's more logical to remember is that only 1.5 million Americans are active in the US Military (not sure why you specified the army specifically other than personal issues). Out of the 120 million Americans eligible to fight, 34% approve of the war (or about 40 million Americans). That means 80 million of them do not. Even if as much as half of the active military does not approve of the war, it's still only less than one percent of that 80 million. Less than 1 percent of those who disapprove of the war and are eligible to fight actually go out and fight it.
What I'm saying is that it was bound to happen simply because of chance. It's a very statistical answer but sometimes the best one to give.