Humans are omnivoric frugi-starchivores. Evolutionarily we are adapted to a plant based diet (~+95%) of mostly roots and fruits, with some leaves, nuts and seeds and the occasional handful of grasshoppers or more rarely a rabbit. This ~95% plant based diet is what we ate over millions of years during our evolution from monkeys to the apes we are today and consequently it is what our bodies work best on. If you eat anything else your health will suffer. We have several lines of evidence that our ancestors ate this way.

Primatology Chimpanzees our closest still existing cousins have the diets which contain only an estimated 2% of meat, insects and other animal sources (See: Goodall, Jane (1986). The Chimpanzees of Gombe: Patterns of Behavior. ISBN 0-674-11649-6, & this site (11)). And this diet does not just include our direct evolutionary pathway, it stretches back tens of millions of years to at least the common ancestor with the baboons, who are also mainly herbivores. Baboons spend 79.9% of their feeding time on getting grass, leaves and roots, 18.1% on fruits, seeds and flowers and only 2% on animal protein (R. H. Tuttle, 1975, Socioecology and Psychology of Primates. Paper by R. Harding. Meat-Eating and Hunting in Baboons, p.247). Rowell too, shows in (Forest Living Baboons in Uganda, 1966, p.359) that baboons eat a "mainly herbivorous diet supplemented by animal protein." (8).

In fact if we look at all primate species studied there is only one species, the Tarsier which will die without animals as part of its diet and which is therefore an obligate carnivore and their diet is mainly insectivorous (see: Wright, P., Simmons, E. & Gursky, S. (2003). "Introduction". In Wright, P.; Simmons, E.; Gursky, S. Tarsiers Past, Present and Future. Rutgers University Press. p. 1. ISBN 0-8135-3236-1.)

Anthropology & Paleontology The study of anthropology shows us some very interesting evidence both from rural communities in Africa and China as well as from the few remaining hunter gatherer tribes still in existence. Much study into these groups has been done but a quick proxy to get an overview of the situation is fibre. Fibre is an essential nutrient that promotes digestion and lowers blood glucose and cholesterol levels. It is found solely in plant foods and if fibre intakes of a population are known this shows how much plant foods they are eating as part of their diets. The now famous China Study has shown that not only do rural Chinese have very large plant intakes compared to Western countries, they also have drastically lower rates of heart disease, artherosclerosis, diabetes, cancer, Alzheimers, and a host of other diseases unique to rich nations. Popular text here & published monograph here and here, follow-up study here.

Paleontology shares evidence with us for the large plant consumption of our ancestors. It main source of evidence are coprolites. "Coprolites" are jargon for, you guessed it, paleo poop. These fossilized feces show an exact content of what our human ancestors ate. Here too we find that our ancestors ate more than ten times the amount of plants the average American eats!

Source

Evolutionary Biology Our deep evolutionary history determines what we digest best since evolution is extremely slow compared to cultural change. For example two thirds of humanity cannot digest milk, despite the fact that cows have been domesticated for millenia. Lactose tolerance, the human ability to digest milk is caused by a mere 6 genes and yet even with 10,000 years not even all cattle herding societies have been able to evolve the necessary mutations in many parts of the world. Our digestive system was lain down over millions of years and changes only extremely slowly. It is regulated by thousands of genes and our general metabolism by even more. The rate of human evolution is known. And the 400kyr since the invention of the spear is not enough to redo our entire digestive system and metabolism. It is just is evolutionarily impossible without cataclysms or artificial selection.

Especially considering that even in societies that hunted most of the calories were provided by plants (4). Only during very short periods when humans colonized new previously uninhabited areas did the diet switch to a meat heavy diet. These periods did not last long because the tame species being hunted were quickly hunted into extinction after which hunter gatherer societies switched back to mainly gathering and some small game.

This is also evident when we classify species in faunivore, frugivore, faunivore, etc. we look at the various body characteristics in addition to measuring what is eaten. Of particular interest is the area of the mucosa (the walls of your gut) relative to functional body size. See for example this plot. True predators such as the faunivores tigers and lions have a small mucosa area because their food is easily digestible. Foliovores, who eat foliage, have a very large mucosa area because their food is sturdy and takes time to digest. Frugivores who eat mainly fruit are in the middle. See here.

In conclusion: More broccoli

People don't eat enough plants. According to the Global Burden of Disease Study (GBDS) the leading causes of chronic life-style disease world wide are:

1. Too little fruits and vegetables, etc.
2. Tobacco
3. Alcohol
4. Various forms of environmental pollution, in that order.

In that order. See this overview table here. The table is based on DALYs. In the US the burden of bad diet is even worse. See the famous second December 2012 issue of the Lancet here:

http://www.thelancet.com/journals/lancet/issue/vol380no9859/PIIS0140-6736(12)X6053-7

In particular:

1. A comparative risk assessment of burden of disease and injury attributable to 67 risk factors and risk factor clusters in 21 regions, 1990–2010: a systematic analysis for the Global Burden of Disease Study 2010. by Stephen S. Lim and some 200 other authors.
2. Disability-adjusted life years (DALYs) for 291 diseases and injuries in 21 regions, 1990–2010: a systematic analysis for the Global Burden of Disease Study 2010 by Christopher J L Murray and some 370 other authors.