Do they come on if you unplug them and then plug them back in?
You seem to be a bit confused how these speakers operate. These speakers have a USB sound card built in to them. All of the normal device communication that would usually happen through the PCI bus is going through the USB cable to a smart sound card device located in the bottom of one of the speakers. That smart sound card device is converting your operating system's sound events into an analog sound signal that is amplified by something. (USB power? Batteries? Logitech's site did not show a power source?) That amplified signal goes to the speakers to make sound.
If the speakers use USB power and the USB port cannot send the requested power AND it knows it, the USB port will turn itself off and notify the OS of an over-amperage. If the USB port gets overdrawn and is NOT smart enough to turn itself off, it will die. (This should not happen in any quality product.) If they don't use USB power (they have batteries or a power brick) then this isn't an issue.
If the speakers come alive when you unplug and re-plug them, you have a driver problem and should look for new drivers.
If the speakers do not come alive when you unplug and re-plug them, you could have a number of problems, and you want to start by making sure those USB ports actually do work. Grab a USB mouse or something and plug it in and make sure it works.
It could even be the USB cable to the speakers. If that's been abused, kinked, tied in a tight knot, or chewed on by rabbits, communication could break down.
You could have the wrong audio device selected as default. Since the speakers essentially install a new sound card in the system, make sure it's set to be the default output device.
If they are battery powered, change the batteries.
Right now I'm leaning towards it being a driver problem so start with the Logitech website.