His arguments are pretty straight-forward: numerous studies have shown that overdosing on vitamins is unhealthy to say the least:
In a study published in The New England Journal of Medicine in 1994, 29,000 Finnish men, all smokers, had been given daily vitamin E, beta carotene, both or a placebo. The study found that those who had taken beta carotene for five to eight years were more likely to die from lung cancer or heart disease.
Two years later the same journal published another study on vitamin supplements. In it, 18,000 people who were at an increased risk of lung cancer because of asbestos exposure or smoking received a combination of vitamin A and beta carotene, or a placebo. Investigators stopped the study when they found that the risk of death from lung cancer for those who took the vitamins was 46 percent higher.
Then, in 2004, a review of 14 randomized trials for the Cochrane Database found that the supplemental vitamins A, C, E and beta carotene, and a mineral, selenium, taken to prevent intestinal cancers, actually increased mortality.
Another review, published in 2005 in the Annals of Internal Medicine, found that in 19 trials of nearly 136,000 people, supplemental vitamin E increased mortality. Also that year, a study of people with vascular disease or diabetes found that vitamin E increased the risk of heart failure. And in 2011, a study published in the Journal of the American Medical Association tied vitamin E supplements to an increased risk of prostate cancer.
He also talks about why this might be (spoiler: too few free radicals might as be problematic as too many), and why there is no regulation against vitamin mega-doses. (http://www.nytimes.com/2013/06/09/opinion/sunday/dont-take-your-vitamins.html)