You have already looked up the verb "to expense" and found the definition "to write off as an expense."
"Headphones that I got expensed" uses the past participle of the verb "to expense" in the passive voice. It can be translated as "Headphones that I got written off as a business expense."
There is a double meaning, or ambiguity, in the verb "got" and the past participle "expensed:"
- These headphones I got [received] were expensed [had been written off] when I acquired them.
- These headphones I got expensed [caused to be written off] after I acquired them.
The first meaning would describe a situation in which Facebook purchased headphones for all their employees and handed them out. The second meaning would be where the employee purchased the headphones with their own money and submitted a receipt to Facebook, and then got reimbursed.
Without more context we can't really say which of these happened, but it is not important to the video. The point is that the employee has the headphones, and it did not cost them money to get them.
Compare both options to the active voice: "I expensed these headphones." This means the person talking was the one who wrote them off. See this humorous New Yorker cartoon:
"The gentleman says, 'You tell me you've got a dastardly plan, then I'll swear to defeat you, and then we can both expense this.' "
Here it is implied that Superman is an independent contractor (presumably being paid by the city for each disaster he prevents, or something) and keeps his own account books.