Supporter frustration at the Riverside Stadium was audible, with former manager Aitor Karanka's name being chanted and fans calling for their own goalkeeper Brad Guzan to be sent off after giving away a penalty.
Bamford, who scored the consolation for Boro, told Sky Sports News: "It's hard but you understand in a way as well because obviously Middlesbrough is such a massive club and it is obviously not nice for anyone that we're getting relegated.
"The fans are entitled to voice their opinion. It is obviously hard. I've been a fan once upon a time when I was young at Nottingham Forest and you get frustrated when you're watching and things aren't going the way you want them to.
"I totally understand. Obviously it is hard as a player but it's one of those things you've got to get on with. We tried to put on a show for the fans and obviously it didn't turn out that way, but we know they'll always be with us."
Boro wrap up their season with a trip to Liverpool next Sunday.