Hi, 'Danny (the producer on the record) didn't have to SWAMP the song UP too much, it was already swamped up pretty good when it came to him' Here's what I think. This is a reference to American Cajun music, which is sometimes said to have a "swampy" sound because the Cajun area of New Orleans has a lot of swamps. The sentence means that the music already sounded very Cajun, so the producer did not have to add much more of the Cajun sound to it.
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The reason it means "clean up" or "clear out" is because it is something that a "swamper" does. See definition 2.a. in the American Heritage Dictionary link here: