Here's an article explaining how MIDI controllers work and what differentiates them from synths.
http://www.howstuffworks.com/midi2.htm
As for playing it without a computer, no it will not make sounds unless it is hooked up to a computer. Think of it as an input device for your computer which acts as the actual MIDI sound synthesizer.
I'd pretty much assume that your grandkids would have regular access to a computer these days since almost everyone does, so they'd still be able to use it after you're gone as long as you installed the software for it. One advantage to using this kind of setup is that you can completely change the sound of the keyboard simply by changing the software you use to drive it.
However I don't really recommend the M-Audio Oxygen line. I use an Oxygen 49 with a laptop at home but I have to say the key build quality is pretty lousy and the only way to be able to play it without it skipping notes is to turn velocity sensitivity off.
Anyway, that pretty much leaves any cheapo Yamaha or Casio as your main choices. They're all pretty much the same though I find the Yamaha's have more natural sounding instruments compared to the cheap and cheerful nintendo sounds of the cheaper Casios.
Offhand my advice would be to simply go to your local big box department store or electronics shop and pick up the cheapest Yamaha keyboard you can find and live with it. 61 notes is pretty much standard and in the lower end of the price ranges there really isn't all that much to differentiate individual models. I'd say anything you can pick up for around 100 UK pounds or so will suit your needs just as well as anything costing two or three times as much.
Gary
