Too many calories is the basic explanation for why American's health sucks. Calories available per person has gone up ~32% since the 1960s (we obviously can't measure calories consumed per person, but supply and demand would dictate these excess calories are going somewhere). It is not clear to me that meat specifically is a problem so much as excess consumption leading to obesity, which then causes chronic health problems downstream.

Though of course "meat" is too vague a category to be helpful. Obviously there's a link between beef and heart disease and colorectal cancer. There seems to be no health problems associated with consuming chicken or seafood.