If humans can live without consuming other animals, then why do we do it? From a biological point of view, why do we eat meat? I would also extend the question to other animals because many animals are carnivorous: why did carnivory evolve and why do animals eat meat?
I'm going to focus on the why do carnivores exist part of this question, which should be extendible to answer why humans eat meat.
Let's start a thought experiment in which we only allow the consumption of vegetation (plants). In the simple system, we have our plants (lettuce) and our herbivores (rabbits). The lettuces have a constant population size and can support a limited population of rabbits. The rabbits satisfy their energetic and nutritional requirements by eating lettuces.
We then introduce a new species (humans) which are capable of eating both lettuce and rabbits (omnivores). If the humans limited themselves to only the lettuces as a food source they would have to compete with the rabbits in order to survive. Alternatively, the humans could hunt rabbits, as rabbits are also a potential source of energy and nutrition. The point being that, if a strategy has the potential to satisfy the energetic and nutritional requirements of a species, then that strategy has the potential to evolve. Because both plants and animals carry energetic and nutritional value, herbivory and carnivory/omnivory can evolve.
Now consider the nutritional and energetic value of two the two food sources, plants and meat. Meat is typically richer in calories than vegetation, particularly those coming from fat and protein. 100 grams of lettuce contains around 14 calories, while 100 grams of beef contains about 200 calories. Protein is used by the body for growth, to repair tissues, produce hormones and enzymes and much more. Fat has a high energy content per gram (roughly twice as many calories per gram than protein or carbohydrate), is essential for the digestion of some vitamins, and is an effective insulator. Given that behaviour takes time and energy, it is perhaps not hard to see that consuming meat can help to satisfy the energy requirements more quickly than consuming plants, allowing meat eaters more time for other essential activity like sleep and reproduction. Consider the activity of two large mammals, elephants and lions. Elephants spend a lot of time eating, some 12-18 hours per day (fortunately for them are quite able to eat on the move). Lions on the other hand typically hunt for short periods of the day (dawn and dusk), allowing them to avoid the heat (reducing water requirements), to rest, and reproduce.
Consider also that vegetation is often hard to digest — it is often is rich in cellulose. Many herbivorous species have complex adaptations to help digestion. Therefore, it's not necessarily easy, or easier, to evolve to eat only vegetation. Improved energy acquisition can evolve by both eating more energy rich foods and by more efficiently extracting energy.
Carnivory is also present in plants. According to this paper , such plants may be an adaptation to nutrient-poor conditions, where the ability to digest insects (and other small animals) would be a massive benefit, allowing carnivory to evolve rapidly.
Botanical carnivory is thought to have evolved in nutrient-poor and well-lit habitats such as bogs because the marginal benefits accruing from carnivory exceed the marginal photosynthetic costs associated with the maintenance of carnivorous organs.
The Egyptian mongoose is an omnivore. This species varies its diet across the year, according to what is available. For example, insects are the major component of its diet in spring, while no fish are eaten during summer (because of dry canals in the summer) and fish are a major component in the winter. This illustrates that allowing omnivory/carnivory can increase the resources available, potentially allowing better survival and reproduction relative to a herbivorous/narrow foraging strategy in the same species. Consider this problem in, for example, boreal or temperate climates, where winter can cover ground vegetation in snow and tree's will reduce foliage. Consuming animals would allow the acquisition of energy and nutrients through the winter.
In summary, carnivory can evolve because animals are a good and plentiful source of energy and nutrition, and may allow diversification of strategies, improving survival and reproduction. If animals were bad food sources then carnivory would be less common as it would be a more difficult/problematic behaviour to evolve.
While many people & organisations (including PETA) try to claim that we are not well adapted to being carnivores, many of the claimed adaptations to herbivory/lack of adaptations to omnivory are overstated, misleadingly presented, and even false, and are used in order to promote their own agenda. It's also worth noting that we are not well adapted to being herbivores (e.g. gut flora) and chimpanzees, one of our closest ancestors, are also omnivores.
 Krebs & Davies Behavioural Ecology, An evolutionary approach.* ISBN-10: 0865427313.
 Ellison, Aaron M., and Nicholas J. Gotelli. "Nitrogen availability alters the expression of carnivory in the northern pitcher plant, Sarracenia purpurea." Proc Natl Acad Sci, USA 99.7 (2002): 4409-4412.
First, just because people can survive on a vegan diet doesn't mean all other animals can function as herbivores. Members of the cat family, for example, have evolved as obligate carnivores.
According to some researchers, the evolution of carnivory is associated with the Cambrian Explosion (Oxygen Brought Earliest Carnivores to Life).
But, as MG_MD suggested, opportunity is an important factor. To put it another way, the answer to the question "Why do carnivores exist" can perhaps be most succinctly answered with the word niche.
If an island is inhabited by just one animal species, that species will obviously be limited to an herbivorous diet (unless it feeds on beach carrion or marine life). But if other animal species are present, then there's suddenly a choice between plant food, animal food or both (omnivory). Stiff competition for plants may "encourage" evolution towards an alternate food source (animal matter). A particular species may also be somehow pre-adapted to carnivory or to catching animal prey. For example, a species that's able to climb trees will have access to the eggs of nesting birds.
Incidentally, carnivores can evolve into herbivores. Examples include the giant panda and palm nut vulture, along with an herbivorous spider.