If you think about what cowboy boots looked like, they had big straps on - hence bootstraps.
Back in the 1800s, the phrase 'pulling yourself up by your own bootstraps' was used to mean something that looked to be impossible - after all, it doesn't matter how hard you pull on your bootstraps, you can't lift yourself up by doing it.
Engineers tended to pick up on this phrase, and used it to mean doing something that looked impossible - i.e. by doing something that you wouldn't immediately expect to work (e.g. feeding the output of an amplifier back into the power-supply, giving a boost in voltage when the amp is under high load, allowing the amp to drive the speakers harder. This idea sounds insane, but does actually work, and the mechanism that diverts power back into the PSU has been called a bootstrap for this reason).
When you think about how a computer and OS work, similar problems crop up. When the OS starts it needs to know how to find and read files off the disk - but the program that finds and reads files from disk is on the disk. So engineers have designed mechanisms that are able to get around this seemingly impossible problem (the computer loads a simple program from the first part of the disk which then loads the program that is able to find and load files off disk). This program that begins loading the OS, has been termed a 'bootstrap loader' because of its similarity to the 'bootstrap' problem above.
With time, bootstrap got shortened to boot.
