Su.Ki. points out an important aspect of the expression that the Chinese translation omits. The reason that you have stuff on your shoe is through some action of yours (you stepped in it) but you probably didn't mean to do it. Nonetheless, there is now a problem that will not be resolved until you do something about it. If you do an incomplete job, there will still be a problem so you must do it completely. And the lack of "disgusting odor" will tell you when you have completed the task.
I don't know the episode and scene that you are referring to. If you'll provide it, I'll watch the episode to see if I can provide more understanding of what was meant by the phrase. Knowing the series, however, I can imagine that the sh*t refers to some political mess or scandal that must be cleaned up or covered up in such a way that it is no longer a problem for the speaker.