It's given away in the title, really – it does what it says – more power with a big bore. A motorcycle's capacity is the amount of space the pistons occupy in the cylinder. This is referred to as its cubic capacity, hence the term 'cc'.
This capacity is worked out from the bore, which is the diameter of the cylinder, and from its stroke, namely, the distance the piston travels from top dead center to bottom dead center. A big bore is simply increasing the width of the cylinder, and therefore increasing the engine's capacity.
Due to the compact construction of modern motorcycle engines, this is becoming something of a dying art, but if you have an older motorcycle, or if your motorcycle is equipped with a single-cylinder engine, then you're well away.
To increase the bore, the cylinder has to come off. Older motorcycles have cast iron liners within the cylinder, inside with the piston runs up and down. Using a honing device, this liner can be increased in diameter, thereby increasing the bore. That's the easy part. Unfortunately for anyone looking to make more power by this route, modern motorcycles don't have this liner. Motorcycles such as Yamaha's YZF-R1 have solid aluminum blocks with plated bores.
That isn't to say that they can't still be honed, just not to such a large degree as older motorcycles with a more traditional construction for the engine internals. Not only that, but they have to be re-plated afterwards, which is expensive.
Once you've got your new larger bore in place, then obviously, you need to make the move to a larger piston to fill the space you've freed up. When all this comes together, you could be looking at up to 40% more power without sacrificing anything in the way of reliability.
Many Suzuki Bandit 1250 owners sing the merits of a big bore as a good way to look for cheap power gains.
At this point, you can go further if you want to, but the costs rise sharply. So far, by simply boring out the cylinder, you haven't changed the engine's stroke. If you really want even more capacity and power from your motorcycle engine, then you have to think about altering the stroke.
This involves swapping con-rods, which can cost over 30,000 THB for the average big bike, and getting the crank out, too. Don't expect to much change from 100,000 THB.