Anthropology is a science of culture, and as a science of culture, it studies people. The four men who began anthropologizing religion in the late nineteenth century (Max Muller, W. Robertson Smith, Edward B. Tylor, and Sir James G. Frazer), made this clear: in anthropology, religion is studied from the outside, not the inside.
Anthropology has nothing to say about whether God, or Zeus, or tree spirits are real, or how they might function in their own realms. Anthropology of religion doesn't aim to explain ghosts, debunk the Shroud of Turin, or investigate the validity of near-death experiences.
Anthropology of religion seeks to unearth the whats and hows, rather than the whys, of religion, leaving the objective verification of religious ideas out of the equation. It focuses instead on learning about cultural practices, norms, and beliefs. Learning about how people practice religion, and their reasons for it, is what is important. Whether or not religious beliefs are true is left up to philosophy, theology and non-anthropological science.
The main way anthropologists learn about religion is by doing fieldwork, investigating cultures directly. While the earliest (Victorian) anthropologists of religion focused largely on foreign, tribal cultures, leaving their own religion and culture out of the study, all religions are now included under anthropology's investigative umbrella.
The field is incredibly diverse. Anthropology is but one of several lenses used in studying humans, anthropology of religion is one of many lenses used to study human cultures, and within anthropology of religion exist even more sub-categorical lenses of study. The anthropological study of religion may view religion existentially (as a creator or mediator of values), by social order (seeing religion as a tool of social organization), or in an affective sense (studying what emotions are associated with religion).
However, anthropologists agree that all religions must have three specific elements: religious ritual, religious ideology (feelings and beliefs that are supernatural), and religious social organization. All three must be considered when investigating the role of religion in a particular culture.
For example, in some of the more supernaturally-infused sects of Buddhism, rituals include meditation and the saying of mantras, ideology can include a heaven-like place called the "Pure Land" as well as the concepts of rebirth and nirvana, and social organization can include the physical or conceptual separation of monks from the laity.
In Judaism, Hanukkah services are ritualistic, ideology is represented in prayer and the stories of the Old Testament, and social organization is influenced (in one way) as only men wear the yarmulke.
The ways in which features like these small examples interact within a religion, creating a complex whole, is of prime interest to an anthropologist of religion. Studies expand to the hundreds of religions on earth, as new cultures and tribal societies are introduced to researchers.
It's hard to argue against the notion that religions should be studied in this way. Religions play powerful, complex roles in most cultures, defining rites of passage, meaningful rituals, moral expectations, and complex modes of communication. Studying cultures without studying religion would leave anthropologists largely in the dark about the whole of humanity.