The big difference between a sandbox game's plots and a more linear game's plot is that plurality. A sandbox game has things going on that have interested parties, of whom either the player characters already are or who view the player characters as possible agents, tools, or otherwise elements of their own plans for the things they're itnerested in.

In the game I reference most often, Jathaan's seafaring game, we started off as a newly-formed company within a mercenary organization, and we had a few jobs we could take at our option. The one we did take was a task to secretly investigate what had happened to a missing ship that rumors or reports its owner had just gotten placed in a particular area. Our captain (of the company) also got some cargo and some "special cargo" to deliver en route, both as a cover for the trip and for extra profits. The "extra cargo" was carried by a mysterious woman who stayed in her quarters on the ship the whole time she was with us, and turned out to be a fey fleeing the pursuit of another fey who wanted an artifact she had. When she absconded with our help under the watchful eye of said fey, she left our payment...not the gems we had expected, but the artifact itself. Which we later parlayed into both help in restoring the lost ship and learning its secrets, and also into some major trades in faerie bargains.

One task we did not pursue that came up a few times was something to do with a sorcerer taking over some isolated area. I understand that plot has progressed to a point that it may become an issue we have to deal with.

Nothing about how we dealt with the fey interrupt was pre-planned plotting by the DM. Only the fact of the interrupt and the nature of the cargo, placing us in the proximity of further plot hooks, some of which we bit on. These, too, were things that were happening with or without our intervention, and our intervention changed how they resolved, got is loot and acclaim, and generally served to make our actions meaningful.

Again, it's not that the GAME had a plot planned out. There were events going on, and the DM had some idea of how they might go if we didn't involve ourselves. Our involvement changed what happened where we were involved, and had ripple effects, possibly, to other events. And where we weren't involved, things progressed either as the DM knew they would, or determined they would as he advanced the timeline of everything.

I don't know how accurate my understanding of the behind-the-scenes stuff here is, but that's my best guess how he ran it. I know for a fact he didn't plan out a plotted arc for us, though, only set up challenges and obstacles and provided motivations and goals in the form of problems to solve or things other NPCs wanted us to do.