What does 짬 좀 차니까 빠져가지고 mean?

See a translation

짬 = 짬밥 = meals you eat in the army
짬밥이 차다 = you've eaten enough meals here = you've been here long enough and becoming fairly experienced. (This expression originates from the army but you can use it in other situations)
빠져가지고 - 빠지다 (in this context) means you are becoming sloppy or careless

So putting the entire sentence together, it means "Now that you think you've been here long enough, you think you can you slack off like this?"
The 'sloppiness' here not only applies to the quality of your work but also to your attitude, especially to your seniors (i.e. you aren't as obedient or scared of them)