According to the Vortaro, "sidi" is intransitive ("To sit"). I saw "sidiĝi" in a duolingo lesson, and I assumed it was a mistake. However, "sidiĝi" is also in the Vortaro, with the definition "iĝi sidanta" ("to become sitting"). To me, "sidi" means "iĝi sidanta". What's the difference, and why is "sidiĝi" a word at all?

Thanks,

Richard

The radiko sid' was defined in the Universala Vortaro (1894) by translations in six languages, all of them intransitive verbs. So the meaning of sidi is "to be in a seated position" (mi sidas antaŭ mia komputilo - I'm sitting at my computer).

Any of the standard suffixes (-et, -eg, -aĉ ...) can be attached to any radiko if the combination makes sense. So if we attach -iĝ and -ig to sid', we get:
sidiĝi - to take a seat (literally: to become seated)
sidigi sin - to seat oneself (literally: to cause oneself to be seated). Compare: Residigu vin! (Get your a** back in that chair!).

With intransitive verbs the use of -iĝ imparts the idea of commencement, so whereas sidi means to be seated, sidiĝi means to sit down, or become seated (as noelekim explained). Is that sense it's very similar to the prefix ek-, and you could use eksidi anywhere where sidiĝi appears more or less interchangeably.

Further info at PMEG here.