To "home-style" something means to do it on your own, in your own way. To not depend on society and rules to help you out.
Joyce: Jim! What happened to your leg?!
Jim: Aw shit, Joyce! My cat bit me and I think the cut may be infected.
Joyce: That's the third time this week. Jim, you outta think about putting that cat down.
Jim: But I don't have the money to put her down.
Joyce: *hands Jim a knife* Then home-style it.
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