Microbiology undergraduate here. Creating a completely sterile environment is pretty much unachievable, so there's always a chance that unwanted microorganisms will attempt to invade your beer or wine. However, by introducing a sufficient amount of the right kind of yeast, they'll reproduce and effectively outcompete (by consuming all available resources) any other organisms, eventually killing them off through the production of alcohol and carbon dioxide.

This next part is somewhat speculative, but I believe that the Romans likely had some basic understanding of hygiene, such as washing their hands using a mixture of vinegar and water.

I'm pretty confident that those responsible for stomping the grapes would have at least rinsed their feet with water before begining the process. It's difficult to imagine that our ancient ancestors would have allowed someone with dirty, contaminated feet to participate in grape stomping. There are numerous examples throughout history of cultures practicing handwashing with water or a water/vinegar solution before meals so it doesnt seem too far-fetched to think that they would do the same for their grape-stomping feet.

I think it is likely that the grape stompers were also the ones working in the vineyards or wineries. Their lives revolved around these areas and interacting with the grapes on a daily basis. After a few days of grape stomping, they would essentially become living, breathing yeast colonies. We're all covered in yeast from head to toe, but these individuals exposure would be heightened to the point where they could kickstart the fermentation of grape juice just by being in close proximity to it. I have read that consuming fermented soybeans means youre not allowed near a brewery, as the microorganisms that ferment can wreak havoc on the sake brewing process.

Yeast colonies don't tolerate competition well, so they work to outcompete other organisms to remain dominant and secure their resources