My fiancé was just told he has low testosterone and elevated liver enzymes and has been very itchy. He is not a drinker. What could this mean?

Various causes. There are several causes of high liver enzymes - viral hepatitis, meds, fatty liver, hemochromatosis. Any chronic illness can cause low testosterone. Hemochromatosis can cause both.
Hepatitis. Elevated liver enzymes suggest hepatitis which is a fancy way of saying inflamed liver. Lots of reasons for hepatitis including alcohol, acetaminophen, other medications, environmental exposures & infections (in past, mainly from blood transfusions, but these days more likely from sexual encounters & dirty needles (tattoos & drugs)). Genetics (hemochromatosis) can do this, too. Go see your doc!