Bowser's Star Reactor is accessible from the Fountain observation dome. You need 15 Stars to enter.
This galaxy is a linear course of platforms in an abstract style.
The Fiery Stronghold
Run past the Fire Bars and up the multicoloured path. Jump up the steps avoiding the Fire Shooter and the Thwomp. A green platform will take you to a section which takes place in a 2D plane. Observe the arrows in the background to see the direction gravity will pull you. Jump up to the red bit at the top, over the Fire Bars, then time a jump to land on the moving platform below. Travel with the platform and jump up at the end, then jump over some Fire Bars and under a Thwomp.
Ride the yellow platform to a stone circle with Fire Bars. Work your way round to the top to a walkway. You'll need to avoid a Thwomp, some Fire Shooters and more Fire Bars. When you reach the green and white checked floor, Bowser will appear and start breathing fireballs at you. These won't hit you but will damage the walkway, which will start collapsing. Quickly follow the path up to the top.
Boss fight - Bowser. Avoid Bowser's jump attack, the blue radiating shock waves, and the fireballs he breathes at you. You can avoid the shock waves by timing your jumps, or by standing on the raised platforms.
To defeat Bowser, you need to make use of the fact that as part of his attack pattern he will try to jump on you, so use that to make him jump onto the circular glass areas. These areas will break, causing Bowser to burn his tail on the lava and start running. While he is burning, spin attack him a couple of times. Repeat the procedure two more times.
Replenish your health with coins trapped in crystal, embedded in the ground (fire Star Bits to release), or hidden in the raised platforms (make Bowser jump on them to release).
When Bowser is defeated, collect the Grand Star, which as well as increasing your Star total will unlock the Kitchen dome.