First, have a look at your reference markings on each side. You will wire each side to the next, connecting all six sides to form a continuous circuit. Figure out which side you would prefer start your circuit on. This is where you will connect the 18/2 lamp wire that will run to the outside of the lantern and ultimately connect to your power adapter. You will want to make sure that this start point is as close as possible to one of the wood spacers. When you are completely done with all of the wiring, you will tack the 18/2 wire to the wood spacer a couple times. That way, if the completed lamp is accidentally picked up by the wire, the wire tacked securely to the wood spacer will bear the stress, instead of all your wiring getting ripped from the inner cube.
Since you picked up 2 feet of each respective color of Flexible LED Strip Lights, you will need to cut each two foot strip in to four 4" sections. Cut only where you see the scissor icons on the strip light.
Draw a continuous line on your inner mdf cube that indicates where you will adhere your Flexible LED Strip Lights (the strip has an adhesive backing). Make sure to think about your starting point based on the notes above.