The nickname is a play on the South Park Episode "ManBearPig", in which Al Gore warns the citizens of South Park about a worldwide danger from a creature that is "half man, half bear, half pig." Because Puig sounds like pig and Yasiel is a big, strong, fast, bear-like human, the nickname is a perfect fit.
ManBearPuig was originally coined for the MSTI Community by VietnamDodger, which inspired Dave to create a T-Shirt design around the idea (see photo). That design subsequently was featured on multiple national baseball blogs, such as Yahoo Sports' Big League Stew and CBS Eye On Baseball. Each of these references gave all credit for the idea to Dave rather than VND, and he did not bother to correct them. Dave is currently using the image as his avatar, along with the handle ManBearDave.