Sandbox games like Minecraft and Terraria have gained a large following over the years and I'm one of those followers.
It's quite hard for me to explain why so many people love these kinds of games. Maybe it's because they are relaxing experiences that the limit is you imagination and let you express yourself in a new way.
I've seen works of art in both of those games and I created some in Terraria. And playing single player makes me roleplay as a survivor in a strange and dangerous world,and I think that's the answer.
And I've started seeing videos of Robocraft and,while not being a sandbox game,I saw the creation of people that I coudn't do anything but awe with the creativity and hard work put into them.
And there's also the survivor sandbox like DayZ,Rust,Unturned,Don't Starve Haven and Hearth.. Simple concept,survive,but really difficult games due to managing several stats,hunger for instance, and because PvP is allowed and you don't know if the next player will kill you or not and that's why it enchances the roleplaying experience.
TL;DR:Sandbox and survival games are popular because you can express yourself and roleplay in a way never seen before in video games.
Now what's your thoughts of sandbox and survival games,their popularity and how they impacted the industry?

Personally I prefer set goals, if I'm being creative I'd prefer to use something like actual Lego for the full tactile experience
But I'm pretty sure that's just how my brain works - enjoying ticking jobs off lists rather than just 'being creative' I'm not saying they're not good 'objectively' whatever THAT means!

Saying that, I DO like open world games - when I played paperboy as a kid and I got to the crossroads where you had to dodge the pink Cadillac I always wished you could make a left or a right and see what was over there so they're like the games I always dreamed of, in a way!

For sandbox games, I tend to be drawn towards those that enable and even occasionally encourage modding.

The Sims 2, as a base game, is a life simulator. Add in a few expansion packs and you can adjust and add to the experience with university, nightlife, hobbies, seasons & weather, and businesses. Thanks to the business expansion, one of the options to run a business was to put down a ticket machine and charge by the hour; the other option was the ability to sell almost anything from the buy mode catalogue (and a fair amount of stuff from build mode such as windows and doors). Thanks to custom content, I can create a ton of businesses based on either a charge-per-hour basis, or by flagging items for sale.

For Skyrim, I've got almost 60 hours in it and I've barely touched the main questline. Huge list of mods in the game, mostly focused around enhancing the game (immersive weapons and armors, cloaks and bags, some new hairstyles). At the moment I'm basically dividing my time between running around and exploring/doing minor quests, and seeing what new weapons and armours pop up and if I can craft any.

For survival games, I can't actually say that much about it, and I've got a couple in my library. Zombies seem to be popular at the moment (The Walking Dead) but even with them removed there's still the challenge and, sometimes, perma-death if you're not careful. I'd presume there's interest in being able to challenge yourself to do better each and every time.

I think you making your own goals is part of the appeal of those games and enchances the roleplaying experience.
Also,I'm thank you for reminding me that I need to buy more Lego this year. I need to complete my Star Wars collection :) WarZ(I'm sorry,Infestation:Survivor Stories) is THE shining example of this.

Running over old ladies and having their brain juice splattered all over my windshield. I'm not joking. Also works for minorities and cops.

I kind of feel that you cant talk about sandbox games without mentioning Saints Row, specially Saints Row three and four (even though SR4 was more of the same, but with super powers). They are just mindless, fun and they dont take themselves too seriously, which honestly is the way I wish the GTA series would have gone.

you mean gta takes it self far too seriously these days ?
gta 4 and its expansion packs sure as hell did

and i have zero interest in gta 5

BTW and slightly OT, the Terraria Christmas change kicked in this morning. I don't know what the hell I'm going to do with all these sugar cookies I keep getting.

Didn't know you played Terraria. And thanks for the update.

That is ALL I play. I have 450 + games in here and spend all my time on just that one. XD

In fact, in response the OP, I wouldn't say I 'love' sandbox games in the same sense that a meth addict doesn't actually 'love' meth. :P

I hinted at it in my original reply reply but I think it's worth trying to draw a dividing line between the related genres of 'Open World' and 'Sandbox' -

Open World is like GTA or Elder Scrolls where you're free to mooch around a big map doing main and side quests as you see fit and the whim takes you

Sandbox is more like Minecraft where it's about creatively building things in your own space and its not so quest orientated - you set your own goals, pretty much. Many old tycoon / god games like Dungeon Master, Theme Park and Rolelrcoaster Tycoon had 'sandbox' modes too - they were similar in that they allowed you to see what you could design when unconstrained by budget, times and other constraints of set scenarios

Obviously there's overlap and games that blur lines like 'Cargo: The Quest for Gravity' where you design craft like Lego to complete set tasks though

Maybe sandbox vs toybox?

Like kids on a playground. There are no rules on how to play. No "Do this now. Now do that." You are free to go anywhere, do anything at any time. It trips people's power centers. People enjoy that freedom to play around however they want, and video games present virtual consequence free environments.

For me in particular? I love to explore. When I look at my favorite video games ever, they are all open world action adventures or RPGs - my absolute favorites are the ones with a focus on art direction and minute attention to detail in the environments. I love surrealism in particular. Zelda, Skyrim, Xenoblade, Myst/Uru, Tomb Raider (2013), and the Arkham series all present fantastic virtual spaces that encourage exploration and sight seeing. I appreciate those creators who put hidden goodies in inconspicuous corners for people to find who really get observant and creative in their exploration. Case in point: the plans for Arkham City that were hidden behind a wall in Arkham Asylum. Rocksteady finally had to give that one away though, but they say there are still things hidden in Arkham City that have yet to be discovered.

A lot of people have fun trying to 'break' the game. Seeing just how far they can push the boundaries. Exploiting glitches and playing with emergent behavior to get exciting stuff to happen that designers had never intended. Players were able to figure out a glitch that allows Batman to glide over to the backdrop of Gotham in Arkham City. Players figured out a unique 'bomb jumping' glitch in Ocarina of Time that allows players to play sequences out of order.

Players also like it when games present an obstacle and then offer players a variety of tools they can use to get around it their own way. Deus Ex is the prime example of a game that lets players figure out their own way through instead of forcing them to figure out what designers want them to do.

tl;dr people enjoy freedom

Sandbox games drive me mad. I need to have some set goal to work towards, otherwise I get bored or frustrated. I used to play the sims when I was younger, but I could never play it for more than a few hours, because everything would become such a drag. With only personal goals to aspire to, I quickly became bored. So I can understand the appeal, but unfortunately, these kinds of games just aren't for me.