I would say that it's less about some kind of absolute truths about existence, which are often things that we have long heard about (be kind to your neighbor, etc) but rather more about the characters and the experiences that they go through in order to arrive at these truths. He strikes me as a person who actually lived through the things that he writes about and he bases his characters and their experiences on them.
Some writers may have some truth that they want to convey in their book and so they create characters that are simply vehicles for these ideas and not much more (ie Ayn Rand). Dostoyevsky's characters are not simply posterboys for some moral agenda but are more like personality archetypes that you would actually come across, that's what makes his writing so relatable and believable, and the interactions between these characters so lifelike.
Some of it does get lost in translation, there's something about Russian dialogue that it translates very awkwardly to English.
PS: Another example where his insight was pretty accurate was his book The Possessed. It was published over 40 years before the communist revolution in Russia but it definitely offered some very on-point commentary about the nature of the people who lead these kinds of movements. It was prescient in many ways, really. It's a lesser known book but I highly recommend it, in many ways I think it's relevant to what's (still) happening in politics today.