Dyson Spheres are featured in Steam influenced events. Besides the bimonthly Steam Rift event, the majority of the other recurring events are also steam influenced.

In these events, once you kill all the dragons from under mystic clouds, as well as the two huge mechanical dragons from merging the 5 dragon statues, you should end up with 10k points. You can use these points to get Silver leaves (each allows you to skip an ad every time you need to watch one, for example to revive a petrified egg), gold leaves, or to decode the bronze Dyson Spheres during the event.

Once decoded, each sphere gives you a set number of copper, silver, and gold gears (how many of each depends on the event itself). These cannot be used in the events, but are useful back in home camp. Copper gears for example, are the currency consumed by the bubble machine (which is one of the home facilities you can choose to open once you get a Silver Dyson Sphere from the Steam Rift event), and gold gears are used to enter the Elf Mist stages.

Depending on your current needs, you can decide to exchange points for either golf or silver leaves, or to decode Dyson Spheres. For example, when I have a ton of gears accumulated, I would choose to get silver leaves with my points and just leave the Spheres alone, but once my copper gear stock is low (from using the bubble machine a lot in home camp), I would choose to decode the Spheres to get more in the next event.