When you make your theory bright (and clear and easy to understand) you must have come up with a great explanation for it. Before, it was dark and murky in everyone's mind. But, when you gave your explanation, it became bright and easy to understand! Yay!

説明 is a general, everyday word that is used when explaining something unknown or unfamiliar to make it understandable. You may see it when explaining how to use a device or describing a situation, for example.

This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.