The idea of a sandbox game is an interesting concept. As opposed to other games, where specific content is given to the player in the forms of rules, goals, and rewards, sandbox games task the player with coming up with all of that on their own. Within reason, there are no guidelines or restrictions that prevent the player from playing the way they want to. Most likely the best example of such a game is Minecraft (2011). Since the release of Minecraft, we've seen an influx of games that focus around the idea of player-defined goals, otherwise known as paidic goals. Examples include Terraria (2011), DayZ (2012), Cube World (2013), Don't Starve (2013), Starbound (2013), and Rust (2013), just to name a few. All of these games in one way or another send off the player to just explore the world and take what they will from it. The idea that players must make their own fun almost sounds like a silly notion, one which players wouldn't buy into. Regardless, these games have still managed to gather healthy followings, especially in the case of Minecraft. The question I'll attempt to address is why these games are so popular and what makes them so attractive to some people.
Let's look at the typical Mario game just to put things into perspective first. Nintendo has taken a liking to making Mario games in the style of Super Mario Bros. (1985) in recent years, so I'll take a look at that particular formula. It's very clear what the player must do to play the game "correctly" and how to go about winning. The concept in these games is fairly simplistic, and very structured, following a point A to point B system. Mario begins at the start of the level (Point A), and must make his way to the end of the level (Point B). This is the main goal for both each level individually, as well as the game as a whole. To beat the game, players must make it from the first level (Point A), to the final level where the final face off with Bowser takes place (Point B). To make traveling from point A to point B more interesting and challenging, hazardous obstacles in the form of enemies, bosses, fire, spikes, and pits are placed throughout each level. As a result of these factors, the main rule becomes to avoid contact with enemies unless you can act correctly to remove them from your path (jump on their head, hit them with a fireball, freeze them, etc.). So in short, players must reach the end of each level, avoiding hazards along the way. This is in essence what comprises a Super Mario Bros.-style game. The reward of these games, the sense of satisfaction and accomplishment players receive, comes from completing the goal laid out by the developers.
In sharp contrast, Minecraft doesn't have such structure. It's premise is completely centered around player-driven goals. Players aren't given a set of instructions, either concrete or implied. Players simply must explore their surroundings and work out what their task is, which they soon realize is anything they want it to be. Minecraft serves very much as a template as it does a game. What I mean by template is that the game lays out a framework for players to add their own content and craft their own stories. For example, dungeons in Minecraft are pretty sparse and small. They aren't styled like typical dungeons in other games which are filled to the brim with enemies and treasure. They tend to feel like maybe they're missing something. This is where mods and adventure maps come in to fill them up and flesh them out. Mods can make these dungeons bigger and more interesting, as well as outright create new dungeons. Adventure map makers can do the same by building onto these dungeons and customizing them into more dynamic places. The dungeons are there for inspiration in a way. They act as a basic outline, like a blank canvas for players to make whatever they want. Minecraft as a whole shares this characteristic.
Mods are a massive part of Minecraft. Mods are probably one main component that keeps the game alive and strong. They add new features such as enemies, biomes, items, weapons, and in some cases they create tangible activities for the player to complete. Adventure maps also play an integral role in Minecraft. They serve to create stories to become invested in and specific goals for players to achieve. Separate from community-created content, players may also set out on their own to make goals within their worlds. Some players might set out to make a massive farmstead filled with hundreds of rows of every crop, perhaps with a large fenced off pasture to the side for their vast collection of animals they spent hours gathering and breeding. Some players may strive to amass wealth in the form of diamond and gold, maybe even to the point where they construct a grand castle out of diamond and gold blocks. Some players may just want to build and nothing else, setting out to construct towns, cities, factories, towers, pirate ships, temples, lairs, and countless other builds. Unlike the Mario game formula where reward comes from a pre-made goal, players feel rewarded from games like Minecraft as a result of developing their own unique goals and then completing that goal. There's a sense of self-fulfillment there, which is stronger than fulfillment gained from completing developer-made goals. This is the case because players feel accomplished for taking the game into their hands and completing tasks they themselves decided to take on.
Funnily enough, even in games with specific goals, players may actually go out of their way to create their own goals, even though it isn't necessary. In Dark Souls (2011), players create their own custom challenges, such as using only a certain weapon, to make the game even harder. This is really important to consider. In a game infamously known for its grueling level of difficulty, there do in fact exist players who desire to make their experience even more challenging. That speaks levels to how players like the idea of creating and completing self-made goals, even when these goals create challenge upon a game that is already challenging. Given that, it's certainly understandable why a game like Minecraft, whose focus is on player-made goals, would be appealing.