눈치 is the skill of reading the situation and how to behave. So the meaning of 눈칫밥을 먹다 is to be in a situation(usually stressful environment) for a certain period so that you gained the skill of reading the situation and how to behave. So for example you can use this idiom if you get in to a company where it is very stressful and you should behave well to the boss.