One of the strongest theories about the origins of 自腹を切る is that it comes from the samurai ritual suicide, also known as “seppuku” or “harakiri.”
Samuri would often commit seppuku before they could be captured by an enemy force. It was considered very brave since it was believed that one’s soul and one’s emotions lived inside the abdomen - and exposing the inside of your own abdomen to all was seen as a courageous act.
自腹を切る acts as a metaphor for seppuku. When we are forced to foot the bill for something our organization or company should be paying, we are sacrificing our own money - and although we don’t experience pain physically, we do feel it psychologically!
自腹を切る itself is a verb, although we omit the を切る part and just say the noun part, 自腹. 自腹 means “one’s own money”, like in the sentence, “自腹で東京に出張した” (“I went to Tokyo for business with my own money”).
I bought the computer for my work at my own expense.
Damn! I lost the receipt! I have to pay out of my own pocket again (although it should be at the company’s expense)!