It's more than just Adult Lego, though.
Many of the adults who love it played back in Alpha and/or Beta when there were no quests, no RPG elements, no goals other than survival (And even that was frequently turned off for something called Creative Mode). It gives you the ability to collaborate with friends (or work alone) to build whatever you want using rudimentary building blocks, and the only thing that stands in your way is your own imagination.
Let me describe it to you this way:
You're dropped into a cool forest-y type world with nothing but your bare hands. You know that in 7 minutes night will come and the zombies will arrive so you have to either find a place to hide or build one. This usually results in your first 3x3x3 "house" where you think you can wait out the 7 minutes of night. But as it turns out, that's pretty boring, so you look down and start digging.
By the time you come up to the surface again to see if it's day time you've been playing for 2.5 hours and can't remember why you were so interested in the surface world in the first place; That cave you discovered is far more interesting.
Eventually you realize you're going to need some proper tools and weapons to survive down there, so you head back up to the surface and punch some trees to gather wood and make tools. Using the tools you make your house bigger and stronger so it can survive a zombie attack and so you have somewhere to keep all your shit.
Now you're 14 hours in and your house is solid, you've explored 500 blocks in every direction, and it's time to build a fucking castle.
6 hours later, building that castle rocked. But it's really lacking a big castle door that closes behind you on your way in, so you put floor plates down on the inside that trigger the doors automatically. Then you wonder if you could control the door with a lever in your inner sanctum instead, so you start reading up on Redstone (basic electrical circuits) and now you've got a door that can be operated from various places inside the castle to let your friends in or out.
About this time you start to wonder. Can I make a trap door? Now you're researching pistons and more complex redstone logic gates to make a trap door that kills zombies when they approach but can be disabled by players for safe passage.
Eventually your friend says, "You know what? I don't understand all this shit you built. I'm gonna go build my own house way over there on that island, and it's gonna have blackjack and hookers!"
Well shit. Now you need to make a minecart rail system so you can go visit him whenever you please.
Anyway, skip ahead to the end, basically you end up hosting a server with 10 of your best friends and before you know it your simple little Minecraft world looks like this.
That is the appeal to Minecraft.