Translation:In the nearest house, the woman sings sweetly.
No, proxima means "next" or "nearest".
"Nearby" means: not far away; close.
Nearest or Next are better, to translate it, because "close" is "prŏpinquus" (prŏpinqua), and proximus is the superlative for propinquus.
propinquuus < proximus
That's true that "propinquus" is both "close" and "next", but using the superlative is even closer. So, if "propinquus" is close, proximus is the closest, the nearest.
They wanted to show us the superlative with multus/plurimus, but don't show it here.
In the Latin texts, the use of the superlative are often blurred with non superlative words, they seems to be often used, not as superlatives, but as normal word intensified, emphasized. But here, as it's a course, we have to make the difference between the normal word (propinquus, same root that appropinquare) and the surperlative.
I translated it with "In the villa, the nearest woman sings sweetly".
I know it makes a lot less sense that their translation, but I think this one is also possible. If they talk about a woman that is close from a particular spot or person. And "next" is also possible, if, for instance, we visit several houses.