Coughlin released 22 players on Saturday to reach the NFL requirement of 53 players on each team's roster ahead of the new season.
He told the club's official website: "It never gets easier, because every year the principals involved are different. My gut starts bothering me about two days before. The day of it I get the headache and the whole bit that goes with it.
"When you speak to these kids and you work with them and you see them, you do see their effort and what they put forth and how important it is to them.
"It's not an easy thing to stand there when a young man walks into his office and as soon as you put your hand out, he starts crying."
New York face the Dallas Cowboys in their regular season opener next Sunday.