Here's the explanation - I would just say what it does in simple terms - Substitute your operating system of choice for Windows here:
The BIOS has control of the computer when it is first turned on before Windows is
loaded. It's job is to initialize the motherboard (i.e. DRAM), perform a few
system checks, and then load Windows off of something. It also tells Windows some things about the computer its running on, and is also involved when the system goes to sleep or is powered off.
Sorry, @Kevin Y, but I disagree with you. "...recognizes and controls various devices that make up the computer," to me anyway, describes more of what the operating system is supposed to do - the BIOS, for example, won't really control PCI addon hardware other than possibly the ability to make it invisible to the operating system.
Don't think it's really necessary to explain it's historical relationship to DOS or anything to a layperson.