Bowser Jr.'s Robot Reactor is accessible from the Terrace observation dome. You need 8 Stars to enter.
This galaxy consists of a couple of small round planets.
Megaleg's Moon
You start on the outside of a small hollow planet, populated with helmeted Goombas and Bullet Bills. You need to use the Bullet Bills to break the glass cage holding the Launch Star. Use the Launch Star to reach the second planet.
Bowser Jr. will appear and set Megaleg onto you, which is a large robot bigger than the planet itself.
Boss Fight - Megaleg. You should climb up one of Megaleg's legs, jump over the Bullet Bills, over the gears, and up to the top of his head. Here you first need to get a Bullet Bill to follow you and break the big cage. You then need to get another Bullet Bill to hit one of the fence panels, which will lower, then get another Bullet Bill inside the fence to break the glass cage holding the Star. If you take too long, the fence panel will raise, and you will have to try again.
If you need to replenish your health, there are coins embedded in the planet's surface, which you can release by firing Star Bits at them. There are also coins in glass cages on the underside of the planet, if you can get a Bullet Bill to follow you that far.