Shame is a big part of Korean (and Japanese) culture. The suicide rate in South Korea is one of the highest in the world, especially for a developed country. Korean bosses also use shame by criticizing their employees loudly and/or in the presence of others.
Agree with @BobPancakes ... isn't 'my mistake' the subject here? I remember in another sentence this exercise they used the particle 에 to indicate what they were being ashamed/disgusted by. Wouldn't 'I feel ashamed of my mistake' be more like '내 실수에 부끄러워요'?