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'Work through your feelings' is a loose term that loosely refers to the process of experiencing an emotional reaction without suppression. Often when people feel strong negative emotion abut something they will try hard to block/stop/suppress experiencing that emotion because it is unpleasant or painful. For example, a person may 'not allow' themselves to cry over a loss. 'Working through your feelings' would involve allowing strong emotion to be experienced until closure is reached (the strong emotion subsides).
I hope this has been of some help. Please let me know if you have further questions or would like me to clarify any part of my answer.