Explanation based on obvious clues in the film (SPOILERS)
Wade is a confident and talented man who smugly looks down on others, acting like a disciple of Nietzsche who has no moral obligation to anyone, least of all God. He senses simple purity in Evans and plays the role of tempter, trying to corrupt Evans and win him to his way of thinking. But as they wait in the hotel room it is Evans who "converts" Wade, in a sense. Evans doesn't accomplish this in any type of contrived manner, he accomplishes it simply by being WHO HE IS -- an undefiled, courageous family man who stubbornly refuses to give up on hope, faith, life, righteousness and family.
As they're waiting, Wade sketches a picture of Evans in a book. It is later shown that this sketch was made on an opening page of a Bible. This reveals that Wade came to view Evans as a Christ figure, at least subconsciously, and that Evans is clearly a type of Christ in the story.
What exactly was it that caused Wade to "convert" and support Evans? (1.) He saw in Evans a man that refused to be bought -- a man who refused to be corrupted by filthy lucre, which was something he never experienced before. (2.) Evan's son insisted that there was still some good in Wade despite his wicked history. Regardless of Wade's denial, the kid was right. The idea that SOMEONE saw a glimmer of good in him, that SOMEONE out there BELIEVED in him despite his past evil deeds ultimately moved him and compelled him to support Evans. (3.) Wade turned evil because he was abandoned as a kid. He saw in Evans and his wife a REAL family -- parents who stayed together and refused to abandon faith, hope, righteousness and family come hell or high water. He became convinced that Evans and his family were worth supporting, even perhaps dying for. (4.) Evans made a deal with the Pinkerton that if he successfully got Wade to the train the Pinkerton would make sure Evans' family got $1000, which would bail them out of their hardships. Wade wanted to make sure the family got that money. Somewhere deep inside he wanted to redeem himself of his wicked past and this was the opportunity. This could be the beginning of a new life.
(5.) Wade's relationship with Evans lasted only a few days but he found something that was missing in all his other relationships - a friendship based on respect rather than familiarity or shared events. Great friendships like this can happen quickly similar to love-at-first-sight. Such relationships are conducive to transparency, which explains why Evans reveals to Wade the real reason he lost his leg in the war.
After Evan's brutal death Wade has an even greater revelation: To see a truly undefiled man -- a simple but GREAT man -- murdered by a group of immoral swine fills him with disgust. You can see it on his face. Yes, he should be thankful that his men remained loyal to him but this was about their only redeeming quality. To truly start a new life he would have to be judge and executioner of the corrupted souls who would pull him back down into the slime.
Why did Evans have to die? Because redemption can only be paid via the spilling of blood. It was the price for Wade to live and have a new life.