Everton suffered a 1-0 home defeat on Sunday, their fourth loss in five Premier League matches, which leaves them 16th after being tipped to challenge for the top four ahead of the season.
Koeman is coming under intense pressure with the Toffees also bottom of their Europa League group, but he believes the club's fans would be wrong to criticise his players' effort.
"At the moment, it is totally different than it was last season," Koeman told Sky Sports after being asked about the supporters' boos at half-time and full-time.
"That's football - it's difficult to take for the players because they gave everything and it's difficult to take for the fans.
"I understand whistles by half-time, I understand everybody being really disappointed about the final result, but they can't complain about the attitude of the players."
Asked if this was his toughest time as a manager after Jeff Hendrick's first-half strike proved decisive, Koeman said: "Being a manager is tough every day.
"This situation always impacts the players, the team and the fans. But I can't complain about the attitude and the commitment of my players, because that was perfect.
"Of course, we can play better on the ball. In this situation it's really disappointing, but a different defeat.
"We will go on, we will continue, because what I saw was the way we need to come out of this difficult situation."
Koeman did not feel it was fair for his team selection, in which he made five changes with Wayne Rooney left on the bench, to be questioned after the loss.
"I don't get these questions in the past until now," he said. "Maybe I won then and you don't get these questions, but if you lose...
"In life you don't get a second opportunity, but I'm not unhappy with what I saw from the team. They showed how we had to play.
"We started the game really well, we played aggressive, we were the better team over the total game.
"But the first shot on target is a goal for Burnley and then you know - if you analyse Burnley - that it gets really difficult. You get a lot of ball possession but they are strong with good organisation defensively.
"Then you need to be at your best, but we lost it in the beginning. We had to score at least one or maybe two goals in the first 20 minutes. Then, in the situation the team is in, you make it more comfortable for yourself, but 1-0 down against the team Burnley is, it is very difficult."