Heart disease. Meats have some transfats, but most are manmade. Transfats are made by hydrogenating oils, which turns them into solids (shortening). Transfats are used in processed foods b/c they last longer on shelf that way -- cookies/pie crusts/chips/popcorn/donuts/creamer/margarine. They raise LDL (bad) and lower HDL (good) cholesterol, which leads to heart disease. Saturated fats are just as bad!
Foreign Biochemistry. All our genes/enzymes which process fats are designed for cis-fats. "cis-fat vs. Trans-fat" refers to the carbon-carbon spatial bond pattern when double bonds between adjacent carbon atoms, are converted to single; lookup in google images. All living cells break double bonds to cis-bonds. Trans-bonds create processing problems for our genes/enzymes which are not adapted to properly handle these.
Cause many illnesses. Most trans fats come from partially hydrogenated oils found in many processed & fried foods like french fries & donuts. Simply put, trans fats are highly toxic.They cause inflammation, raise cholesterol & may cause up to 100, 000 deaths in the us from heart disease each year. They also increase risk of alzheimer's disease, diabetes, obesity, depression & infertility. See my comment for some links.