You'll need a 24v DC power adapter for that PSU. To use batteries you'd need a pile of high quality 18650s or bigger and the associated charger(s). I'm not sure how much power that will draw but I have the 12v version and it draws just over 2.1A. In theory the 24v version should draw half as much but you can always email the guys at MI and ask them the current draw so you know what to get. You'll need to connect the red wire to the +24v DC and the yellow (enable line) and black (negative) to the negative side of the power supply.
edit: You'll enjoy that laser when you get it up and running. The large tube in my avatar is the same tube I bought from them a little over a year ago. They should also include the needed ballast resistor along with the tube but if they don't *DO NOT* run the tube. You can ruin it quite fast by doing that.