Learn French: Je regarde le voisin dont la nièce est très sociable. - I am looking at the neighbor whose niece is very sociable.

In French, when you use dont to mean "whose", you still must include the definite article (le, la, or les) before the noun that is possessed. Even though we say "whose niece" in English, French literally structures it as "of whom the niece" (dont la nièce).

How exactly does dont mean "whose" in this sentence?

Dont replaces a noun introduced by the preposition de (of). If you split this into two separate sentences, they would be "Je regarde le voisin" and "La nièce du (de + le) voisin est très sociable." Dont replaces that "de + neighbor" concept, linking the ideas like "of whom" or "whose" does in English.

Why isn't there a word for "at" after regarde?

The French verb regarder translates to "to look at" or "to watch." The preposition "at" is already built into the meaning of the verb itself, so you never add an extra preposition like à after it. You just attach the object directly: Je regarde le voisin.