This is a phrase I’ve borrowed from so-called “heathens,” a word often used for modern norse or germanic pagans by that community. I don't personally like the term heathen, which doesn’t really matter because I focus more on Greco-Roman traditions, but I admire the tendency in the heathen community to place a strong emphasis on morality and community. For heathens, this phrase “gifting cycle” refers not just to exchange between man and the gods, but also exchange among men. In their system, the exchange of both physical and intangible gifts such as food, affection, support, wealth, and aid is the process which is responsible for creating the bonds between people that make community and communal life possible. This practice is rooted in the Indo-European traditions, and has equivalents in nearly every Indo-European tradition we know of.
In the same way that bonds between people create a human community, bonds between the Gods and the primordial substance of being creates the cosmos. That might not be an adequately precise way to word it, as “substance” and “being” are pretty loaded terms philosophically and not all traditions hold that there is any sort of “being-ness” that isn’t either a God, part of a God, or created by a God. Endless arguments can be had over this sort of thing, but we’re going to learn to crawl before we can run and basically limit our explanation to “the Gods do it.” If we’re successful, then someday more virtuous people than us can argue about this all night while reclining next to a vestal hearth. The important thing, at any rate, is that the Gods are engaged in a process of offering their own affection, talents, and substance to the world, to each other, and to us. Emulating that process with ourselves, our community, and our environment is therefore an important part of becoming like the gods and living rightly.
Just as we form bonds with humans by investing time, effort, and material, when we invest time, effort, or material into learning about the gods and acting righteously we transform our relationship to the Gods. They give many gifts to us constantly and thanklessly; life, happiness, and love are all given even to people with no consciously cultivated relationships to any gods (though it could be argued that these are all diminished, and especially so when atheism occurs at a societal rather than individual scale). When we dedicate our own substance to the Gods by learning, acting, and offering worship, we change from being passive recipients of gifts we don’t understand or appreciate into being affirmative and willing participants in a cycle of exchange which improves our lives and our selves. This is not a matter of “subjecting” ourselves to the Gods as some neopagans might complain; that subjection occurred the moment we were created. Participating willingly in the gifting process is a step away from slavery, not towards it, because through it we begin to share the sovereignty of the Gods over our own lives rather than letting it occur unbeknownst to us.
When polytheists use images and statues in worship, it is understood that these icons serve as symbolic objects that help to anchor our physical, animal selves to our worship by granting our eyes an image to associate with the incorporeal forces that are being engaged. Similarly, when we offer food, drink, or votive objects like weapons or jewelry, what is being exchanged is not the physical being of those items but rather the powers that are symbolically vested in them; food is strength and sustenance, drink can be purity, merriment, or revelation depending on if it’s water or alcohol, and various other objects carry different meanings.
The use of those items serves a similar purpose to the use of an icon: It forms mental connections between the motions of the physical body and spiritual phenomena. Our stomachs already know firsthand of the nourishing power of food, and so when we symbolically return that nourishing force to the Gods through offerings the action is understood in a more immanently holistic way. In a sense, we are sublimating our bonds with the experiences of hunger, fulfillment, purification, strength, or whatever else is vested in the offered items, into a bond with the worshipped God. That bond already existed and would exist regardless of practice simply because of the nature of the Gods and our position relative to them, but the ritual process of creating a way for our minds and bodies to perceive and contribute to that bond is perhaps the first and most vital step in establishing religious practice. All subsequent religious engagement happens as a part of the gifting cycle. The process can wax and wane as life goes on, but it is always the basis of our relationships, and relationships are the basis of existence both for man as a social animal and for the cosmos as a series of bonds between different forces and Gods.