Which particular laser did you buy? The power supply on that doesn't look like any of the ones in the photos of other RGB lasers Newgazer sells. It also looks like you photographed that in a mirror right? That says -12V+, not -15V+ if I'm not mistaken. The red jacks are the power jacks, and the blue are the modulation (on/off).
You need to provide each laser driver's power supply (red terminals) just 12V -- not the +/- 15V Aleksa mentioned. The modulation inputs (blue terminals) need about 5V (usually as little as 3V) to turn on each of the lasers. Those 9V batteries won't do it because they're literally constructed from 6x AAAA batteries (4A) in series. They can't provide much current and are only good for stuff like smoke alarms. You're also undervolting the power supplies by about 3V, which reduces the output power as well.
If you're serious about trying to power up your laser, what you need to do is find a 12V power supply that can provide a decent amount of current. You can get
something like this to provide 12V and enough current. That power supply in particular has a nice screw terminal jack for attaching wires to. If you don't want a power supply like that, you can sometimes find large 6V primary cells at electronic stores and string them together for 12V. You can also use RC battery packs. You just need 12V and batteries that can supply enough current.
It looks like the wires they provided you have plugs on each end. For this you'll need to have a plug on one end that you plug into the board of your laser, and exposed wires at the end. You'll need some wire strippers for that. If you don't have any, you really ought to learn some basic electronics/soldering skills before potentially damaging your $230 laser.
Assuming you can get your wires stripped and such, you should first plug in the three modulation cords into the blue sockets, and make sure each respective pair of wires is connected together (shorted). This ensures that your laser are "off."
For power, first make sure your power supply is unplugged. You now plug in each of the power wired connectors that came with your laser to the red terminals each laser's power supply. Then you wire the + side of each of the power connectors to the + of the power supply jack. Likewise, the same with the - wires. It'd be best if you could solder these wires together instead of just twisting them together. You can probably fit each bundle of wires into the screw terminals though.
For the modulation inputs keep them shorted (connected) until you're ready to fire the laser. Make sure that your laser is firmly mounted, and the output of the laser is aiming at a non-reflective surface. Goggles would be good here, but your RGB laser has all the colors of the rainbow. It's more important to be prepared and be careful about what you're doing.
When you're ready to fire the laser, connect power and then disconnect the two modulation wires. This may cause the lasers to light up because the voltage will "float" above the threshold value. If not, you need to put about 3-5V between the + and - leads of the modulation inputs.
Make sure to take more pictures if you're stuck. Also if you don't understand anything above, you need to go learn more electronics. Then you can get stuff like this working.