"The family sat down at the table."
Translation:La familio sidiĝis apud la tablo.
As a native speaker of English, "at the table" is where I've always sat to eat, yes, if that's what you're asking. If you're "at the table," you're properly seated with your feet under the table, all properly and politely ready to eat, or eating.
If you are "next to" the table, you aren't sitting properly to eat. You might be standing, you might be playing with a dog or a toy on the floor, or you might be sitting near enough to talk to someone who is "at the table."
Remember, prepositions are just strange and arbitrary. They never match exactly from language to language.
Because verbs like sidi, kuŝi, stari etc represent the stationary acts and not the actions required to come to that positions. This is a distinction Esperanto makes but some other languages like English doesn't. "La familio sidis" should be understood as "the family was [already] sitting" if you like. By saying sidiĝis, you indicate that they were standing before and they transformed their bodies into the sitting form to eat :).
Another example: "Mia patro eniris en la ĉambron kaj sidiĝis". Saying sidis would not work as nicely, because sitting down from staring is different than stationarily sitting, and what is meant here is that my father came and sat down, he was not already stationarily sitting.
The family is in the act of sitting at the table, implying direction. In another sentence I encountered in this lesson, "Sit down on this chair / Sidiĝu sur ĉi tiun seĝon", we clearly have an accusative. In both cases, direction is implied, so why is this "apud la tablo" instead of "apud la tablon"?