Firmware is like the BIOS for your computer. It holds all the information a device needs to function. Firmware is not only used for DVD burners but is used on a wide variety of devices. But, we're talking about firmware for optical drives here so I will concentrate on answering that
To flash a drive means to upgrade its firmware. The term flash comes from the chips in the drive on which the firmware is stored. This memory is called flash memory because new information can be stored on it, overwriting the old information. How one flashes an optical drive depends on the manufacturer of the drive. For most optical drives you can download an executable file which you can start in Windows and will upgrade the firmware of your drive. Sometimes though you can only download binary flash files and you need an extra application that actually loads the binary firmware file and transfers it to your optical drive. Flashing firmware is not OS dependant but I can imagine not all manufacturers have flash software that supports, let's say Linux.
Why should one flash a drive? Generally speaking, newer firmware versions add more support to the drive. In case of a DVD burner, a newer firmware version may add more media support or better media support. Sometimes newer firmware versions also add new features (such as the ability to burn Double Layer discs). Generally speaking, the newer your firmware version, the better your drive will perform. Sadly, this is not always the case and some people will say "if it ain't broke, don't fix it". Personally, I always install a new firmware version when it's released and I haven't had any bad experiences with that.. so far
Hope that answers your question!