This is very much not an online thing, thus, this is the wrong forum...
However, to answer your question; Burn-In Reduction is a tool that forces your Wii to display what I suppose you could call, a screen saver. This is mainly for use with Plasma and LCD displays. A Burn-In occurs when your TV set is displaying the same, static image, for a long period of time. What this screen saver does, is basically to either dim the output from the Wii when you're not using it, or to display a moving image of some-sort that prevent any one image from being burnt into your screen.
I am assuming with the Wii, that the output dimming is how it handles this. If you have a Plasma TV, or an LCD, then this option should definitely be on. Even if you don't, there is no reason why it shouldn't. In fact, I have no idea why it is even an option, and why it just isn't on all the time.
Well, hope that helps. Hope I didn't confuse you. I went in circles a bit. You probably get it.