The Darwinian would say, "Gene dissemination: Successful competition in the hunt allows the fox to survive and breed, maximizing its individual representation of genes in the population. That's how natural selection works."
The ecological biologist would say, "Sure, but first and foremost, the fox is eating the quail to stay alive. It's a cross-species energy transaction between two individual members of a tightly integrated ecosystem. As a result, the fox survives and manages to pass genes on to future generations. Natural selection is a side effect, fallout from the essential process of seeking energy to stay alive."
A systematist, mainly concerned with phylogeny (evolutionary genealogy), would say the fox is eating the quail because that behavior represents an ancient evolutionary adaptation. A cladisticist (a more narrowly construed systematist) might beg the question by insisting that eating was irrelevant here, but the teeth revealed significant canid relationships. A developmental biologist would ...