I believe it stems from the phrase to "pull yourself up by the bootstraps", which is an old phrase meaning "to succeed without external help", for example "My dad built his business up all by himself! He had to pull himself up by his bootstraps!"
In some really old computers, there was no BIOS equivalent. You literally had to put bits into memory using switches before starting the main processor.
Computers with boot code in some form of ROM could do their initial power-up without intervention from the user. The code that is called when you immediately turn it on was known as "bootstrap code", after the phrase. The machine is brought up by the bootstrap!
