Boot sector malware spreads by modifying the program that enables your computer to start up.
When you turn on a computer, the hardware looks for the boot sector program, which is usually on the hard disk (but can be on a floppy disk or CD), and runs it. This program then loads the rest of the operating system into memory.
Boot sector malware replaces the original boot sector with its own, modified version (and usually hides the original somewhere else on the hard disk). The next time you start up, the infected boot sector is used and the malware becomes live.
You can only become infected if you boot up your computer from an infected disk (e.g. a floppy disk that has an infected boot sector).
Boot sector malware is rare toady, although more recent examples include Mebroot also known as Sinowal, a password-stealing Trojan for the Windows platform.