Here's how it breaks down:
* Nagsunog: means "burned"
* Kilay: means "eyebrows"
The image it evokes is of someone so intently focused on their task that they are literally not paying attention to anything else, even to the point of forgetting to blink and potentially burning their eyebrows!
Example:
"Nagsunog ng kilay si Juan sa pag-aaral para sa exam." (Juan was burning the midnight oil studying for the exam.)
In essence, "nagsunog ng kilay" means someone is putting in a lot of effort and dedication to something.