Carlo Ancelotti's men suffered their second loss in three La Liga matches as Arda Turan scored a 76th-minute winner at The Santiago Bernabeu.
Casillas was booed and whistled at, but said the only way to react was to work hard and perform well.
"The crowd have the final say and if they think they should whistle you have to accept it and respect their opinions," he told reporters.
"The way to respond is to train and play football."
Tiago headed Atletico into a 10th-minute lead before Cristiano Ronaldo won and converted a penalty to level.
But Turan struck the winner, curling Juanfran's cutback into the bottom corner after Raul Garcia's clever dummy.
Casillas felt his team deserved a draw, but praised Atletico for their ability to grind out a result.
"The second half was a blow, we were controlling Atletico well," he said.
"We were taken by surprise on a corner and Turan struck brilliantly. The result is a bit unfair and a draw would have been right.
"But Atletico know how to work their games and we have to congratulate them, just as on other occasions it's us that wins."