The speaker and his co-worker were talking about a situation that they didn’t need to talk about anymore. In order to signal it was time to wrap things up the speaker said “Let’s put this horse to bed.” This is a nice congruent conflation of “put (something) to bed” and “put a horse out to pasture”, meaning to finish or retire something. Perhaps the speaker dredged up in his mind the Godfather scene with the horse head in bed. That certainly finalized things. A big thanks to Joel for actually unintentionally uttering this one and sending it in.