The batteriser is a small device that can be attached to an AA battery, and which taps into the extra charge left over after the voltage has dropped below the cutoff threshold of whatever device you were using the battery in, thus prolonging its life. It does this by increasing the voltage of the battery back to 1.5V (using a voltage increaser circuit). The site claims that this can extend the battery life to up to 8 times.
To understand the claim, consider this typical battery discharge curve:
As the battery is drained, its voltage decreases. Battery-powered devices have a minimum voltage that they require to operate, which is the cut-off voltage. When the battery charge drops below that voltage, the device can't work anymore.
The website claims that the typical cutoff voltage for most everyday devices is between 1.35 and 1.4 volts per battery. If you look eg. at the blue curve above, you'll notice that after the battery has dropped below that amount, there's still quite a large amount of charge left in the battery, if only it could be used at the required voltage.
This sounds all good and dandy. The problem? The 1.35-1.4 volt cut-off claim is a lie.
Most battery-powered everyday consumer electronics, including things like remote controls, game controllers, wireless mice and keyboards, etc. have a cut-off point between 1.0 and 1.1 volts per battery. This has been repeatedly tested by afficionados and professionals. (And, in fact, these devices are explicitly designed with that cut-off voltage precisely because they maximize the battery life of the typical AA battery, most of which have a discharge curve similar to the above). Even the most power-hungry (or low-quality) devices have a cut-off point of at most 1.2 volts.
If you look at the curve again, and see how long the device will work with a cut-off point of eg. 1 volt, you'll see that it will have used well over 90% of the battery charge before dying. A 1.1 volt cut-off point is able to use somewhat less, but not significantly (perhaps in the 80-90% range.)
So, the "batteriser" may do what it claims to do (ie. increase the voltage of the battery to 1.5V regardless of its charge), but it won't actually extend its life in any significant way. The fact is that the 1.35-1.4 volts figure has been deliberately chosen by the marketing of the device in a deceptive and dishonest way. The figure is incorrect for the vast majority of battery-powered consumer products. (It might be true for some very rare, probably very low-quality products, but it's not true for normal devices.)
There may also be drawbacks in using the device: Granted, the website recommends using the device only after the battery has been drained and the device doesn't work. However, if you were to use it from the get-go, it would actually shorten the life of the battery, not increase it. That's because the device itself consumes some of the energy (it's physically impossible to get a 100% efficiency on any physical electric device.) And if the battery was already going to be used 80-90% (or even more), adding this extra baggage is only going to drain it faster.
More damningly, however, the "batteriser" may actually short-circuit the battery due to its design. The entire jacket of a battery is its positive terminal (and the bottom cap is its negative terminal, insulated from the positive one by a thin insulating ring). The "batteriser" is metallic, and if any part of it makes contact with the jacket of the battery, it might short-circuit it. Short-circuiting a battery can be dangerous (for many reasons, all the way up to causing a fire).