To trigger the fight with The Radiance, you must go into the fight with The Hollow Knight with Hornet outside the temple. When fighting the Hollow Knight she'll attack him and hold him down for you to hit him. Instead of attacking the Hollow Knight, hit him with your Dream Nail. You'll be transported to the home of the Radiance. If you should die (you will), you'll have to fight the Hollow Knight every time to get to The Radiance.
Good luck.