Brosseau pitching for MIL—backstory?

Could someone educate me about why MIL would have a position player pitching? Brosseau didn’t look too happy to be pitching; in fact, he seemed embarrassed (perhaps because he knew he was so bad at it?). Also, Bagwell mentioned something about a run rule applying, but it wasn’t clear to me how this related to Brosseau’s pitching. I’d appreciate some education here. TIA!