"A rainbow runs through it: colourful camera tricks – in pictures"
Used in the article's title, "rainbow runs through it" is merely a reference to the "Rainbow Rig" in the second sentence you copied.
"I set a long exposure and run through the frame carrying this Rainbow Rig"
In this case, to "run through the frame" means moving from one part of the camera frame to another. As you can see in the article, the path the writer took with the Rainbow (lighting) Rig adds an artistic element to the composition. That path somewhat resembles a river - and this is the meaning Kris explained.
A side note, there's a movie called "A River Runs Through It" the title of which is a multi-level reference to the story's location, the events of a group of individuals and their lives together (like streams connecting to other streams and rivers), and a reference to the river-like journey of the main character's life from his late teens to his death.