Please explain the notion of ontology as an alternative to culture within Anthropology.

Culture, as we know it today, has been defined for us from an epistemological perspective: what we know about reality (or truth) can be realized either through empiricism (experience) or through rationalism (reason). From an epistemological perspective, culture is what we know of the world, where there is only one human reality supporting a plethora of worldviews.
The ontological turn to anthropology, on the other hand, favors the perspective that there are many worlds and multiple realities (beyond that of human reality). The ontological strategy is an effort to approach these other realities with an open mind. In other words, the epistemological approach favoring either empiricism or rationalism (culture as defined by humans) can be invalidated in the search to embrace new realities.
Consider that in How Forests Think: Toward an Anthropology Beyond the Human, Eduardo Kohn proposes the idea that the native Amazonian Runa people should engage with the sylvan realities within their midst, namely the spirits, animals, and ghosts that dwell in the tropical forests. The ontological approach basically seeks to readjust our human perception of reality to include non-human perspectives and analyses.
In Eduardo Viveiros de Castro's ontological approach to anthropology, he aims to encourage scientists to embrace the Amerindian concept of perspectivism, where reality is based on multiple concepts, and all are equally permissible. For example, he claims that manioc beer is to humans what blood is to jaguars. In other words, the jaguar does not 'believe' that blood represents manioc beer to them, but rather, blood is beer to them. For more on the ontological approach to anthropology, please refer to the links below.
https://www.academia.edu/6292081/The_Ontological_Turn