No one can really explain this game. It's a sandbox that does all sorts of things that has many different ways to approach it.

Not just any sandbox, this game is what I'd call the greatest sandbox game in history. It's like the Velvet Underground of video games, the first 1000 or so people that played it all became video game developers, and many of them made gigantically successful games from casual town builders to other sandbox games like Minecraft; whole new genres spawned. The game obviously has a lot of influence from Peter Molyneux and merged stuff from Will Wright → who made SimCity, the Sims, SimEarth, SimAnt, and so on, and takes elements from ALL of these games. It's also the game that redefined crafting.

It's a sandbox game which is the first one that adequately developed emergent storytelling (the sort of storytelling that emerges when playing tabletop and other live roleplaying games) into a video game form... 

Most people who know anything about the game know it from its fortress mode, and brush over some of the other elements. But the game is also a strategy game, a game that simulates weather patterns, geology, and biospheres. This in turn impacts the sort of wildlife and empires that grow across its surface. You more or less start as a single city in a single Dwarven Kingdom. And you can expand it, making your city more prominent in your Kingdom, and helping your Kingdom to dominate. You can send raids/attacks toward enemies. And there is even an adventure mode where you can put together a party and adventure through your world... this is a less popular mode, but it shouldn't be.

This game has been available for over 20 years, being available to players since 2006, and only recently had its first official Steam release... although the free (under development) version of the game is still available, and will continue development... possibly until the creators die.

So let me try to explain:

It's like SimEarth: because it deals with an ever moving world, expanding biomes, and such.

It's like Oregon Trail: because you gather your supplies and embark on an adventure to start your city, and how you start will shape your early settlement flavour.

It's like SimAnt: because you control a small group of characters, and they expand and build an underground mansion... although you can build towers and castles topside, so it's more like SimAnt time 25,000

It's like SimTower meets Sims: because each character has needs and such, but it goes way deeper, there are lots of different elements that build a Dwarf's psychological profile. Also, past events impact your dwarves, their house history, their friendships, and so on. So Sims on steroids.

It's like Victoria 2/3 and Minecraft: Because you craft and manufacture components and goods from raw materials using certain machines, that can then be further manufactured into even higher goods.

It's like AD&D and Crusader Kings 2/3: because there is a ton of emergent storytelling, just that the game generates it from activities that go on, and it's widespread, you can pause the game and spend up to hours to explore the stories of everything happening concurrently. So, much wider than AD&D.

It's like SimTower SimCity: you build an advanced city, with specialization.

It's like a Rogue: because you can adventure through your generated world.

It's like Lord of the Rings (book)... really Tolkien's Middle Earth in general: Because there are monsters of terrible strength buried in the deep places, and dwarves anywhere in the world might unlock these, and they can cause chaos across the world. Also, a Necromancer might threaten to take over the world. Orcs, Elves, Humans, and all that jazz. Also, different ages of the world based on its contents... for example, if you kill off most of these ancient beasts, and the civilizations of the world take over, then the age will change to reflect that historical difference in your world.

The game is also like Jenga: eventually your fortress will collapse, it's just a matter of what sort of disaster occurs, because the world is chaotic, and at times various contradictions will occur, a single troll (like an internet troll) in your colony might sew dissent and unhappiness that spreads and eventually turns into a riot that wrecks you... Your wealth, and everything you've built will become exposed to raids, and eventually it's a cascade of catastrophe. So games like Don't Starve used the idea of "your game will eventually end in disaster" but I don't liken Dwarf Fortress to that, because Don't Starve is almost purely by running out of resources or outside factors that are scripted... Dwarf Fortress makes you the architect of your own demise. Much like other games, you can play on some kind of baby-mode and never have this happen... personally I have no fun doing this, I love the Jenga aspect.

Here's someone else explaining why it's great:

And you'll probably never find a video the adequately describes the game, but here's one that tries to explain it:

Here's someone else explaining it in relation to another game called "Rimworld" that is based on a part of Dwarf Fortress... This one might be the best explanation.

This video series explains why Dwarf Fortress isn't actually as difficult to play as its reputation, it's just a matter of directing your attention to certain elements. This guy in 2010 had the best tutorial video series. It's out of date, but the best I've found... Most DF videos are far too brief and don't cover nearly enough, or far too long because they take forever to get to the point. This series is comprehensive, to the point, but takes a while to get through it, but despite it being out of date the concepts are still the same now as 2010... Just things are probably easier, and in the Steam version the interface is far more consistent. So look at this as more of a conceptual and "how to approach the game" rather than a precise guide on how to navigate the UI and content:

(Note: This is kind of a rough draft post, I'll fix it up later with structure, grammatical correctness and shit... but I think there's enough here to start a conversation)

I tried Dwarf Fortress what must now be a long time ago, and I just couldn't dedicate the time it required to learn it sufficiently. It seems that the game is much more approachable with the Steam release and a proper graphical user interface, but at the same time, it probably still requires a good bit of time to learn, and the more approachable version also costs money and seems to have pretty poor discounts, so I'm waiting for the Perfect Moment (which might or might not exist) to get the game. I'm pretty sure I'm going to like the game once I dedicate enough time to learn it, but it's not easy to find a suitable moment for that.

Regardless, Dwarf Fortress certainly has a reputation, and probably for a really good reason too. I have no doubt it's something really special and probably deserves way more attention than it currently receives.

I admire the sheer complexity and freeform interactivity the game achieves, but for me personally, I just can't get into it due to the rudimentary graphics.

I need more visual feedback than this to really engage with a game.