Okay, so I should disclaimer the fact that everything I'm saying is pure speculation here. I don't know anything for certain, and this is the only video I've seen where the spin game is in action.
Toy spins are earned, although I don't know if that's exclusively in the playsets or if they can be earned in the toy box as well (I'd guess the latter). I don't know if they're tied to the number of stars you earn or if they're seperate. In Toy Story 3, stars were earned by completing missions, both story missions and side missions. Not sure how it works in Infinity.
The red, yellow, and green icons you can choose from in that video all have different symbols on. It's hard to tell, but I think the yellow one has either a cube or a hexagon, which is the same shape as the power discs that contain items (rather than the round power discs that contain power-ups). In the menu associated with it, I can make out the stunt car, Scrooge's money vault, Spaceship Earth, possibly Cinderella's castle, the boom box, and some terrain pieces. So it could be that the yellow symbol means items that are basic toys, like scenery pieces and mounts. The red symbol has a dotted line and an X symbol (or plus symbol, I'm not sure). This could maybe represent a different catagory of toys, such as race track and rail pieces. Or maybe logic pieces, the ones that can connect to other toys to create a function. The green has a question mark in a pyramid, so maybe that could be the logic toy symbol, or it could represent a random toy from any category.
So, yeah, all speculation, but I'm guessing there are different toys categories to choose from.
Within those categories, it seems like the spin function operates like a fruit machine or a luck wheel. You spin, you scroll through those five sub-categories, and it appears you can stop the wheel to try and get a specific toy. (If you don't stop it, it may potentially slow to a stop on its own.) Of the sub-categories of the yellow symbol shown in the video, each has its own symbol listed in the hexagon above that category. The centre one appears to be a mountain, which matches with the fact that the pieces are all terrain-based. I can't make out the second, and the first (which appears to house mounts and possibly NPC blockheads, it's hard to make out) looks like a star. Although I'm squinting to see it, so I could very easily be wrong. The fourth category, which we don't see the symbol for, looks like it's buildings and similar pieces (as well as Spaceship Earth and Cinderella's castle, there's the flaming ring and what looks like stadium pieces). The fifth is possibly scenery, since I can see a lot of trees.
So that's all just guesswork, and I could easily be wrong. It's just what I could deduce from watching the video. If anyone else knows more, I'd love to hear it, since I'm going to be obsessively spinning this thing every chance I get!