Measure & Cut Wires
Let's start by getting four pieces of wire. I recommend using 26AWG silicone coated wires. These should be about 75mm in length. They don't have to be multi-colored but it's nice if you have them to better decipher the connections.
Strip & Tin Wires
It's good practice to tin your wires before soldering them to pins - this will help prevent the strands of wire from fraying.
Use a pair of wire stripper to remove 5mm of insulation from the tip of each wire. I like to secure all my wires to a helping third helper and tin them in a group. This is much more convenient than setting up each one.
Bundle Wire Set
With all four wires stripped and tinned, lets bundle them up using a piece of heat shrink tubing. This stuff is normally used for insulating exposed connections, but I like to use it to group wires togeher. This will keep the wiring nice and tidy.
Install LED Display to Backpack
OK, now attach the 7-segment LED display to the backpack PCB. Be sure to insert the display in the correct orientation. The PCB has a outline and markings, use them to determine the correct orientation. The dots on the PCB should match with the dots on the display. Line up the pins on the display and carefully insert them into the pins on the PCB. If the pins don't fit, you may need to slightly bend the pins into place.
Solder LED Display to Backpack
OK, now it's time to solder the LED display to the backpack PCB. I like to use a piece of mounting tack to keep the display in place while I solder up the pins. Alternatively, a piece of tape is suffice. I suggest using a piar of helping third hands or a panavise jr. to keep the PCB steady while soldering.
Trim Excess Leads
Inspect your solder joints and ensure you've applied a sufficient amount of solder to the pins. Make sure there are no cold solder joints. They should look like Hershey's Kisses.
Once they're good, you should remove the excess pins from the LED display. Cut all them short using a pair of flush snips.
Solder Wires to LED Display Backpack
Now it's time to connect our four wires to the pins on the Backpack PCB. I suggest tinning the four pins, located on the top of the Backpack PCB first.
The order of the colored wires doesn't matter too much but I like have some sort of sense. Red is positive, Blue is negative. Serial clock (SCL) and serial data (SDA) can be whatever color you'd like. The colors are just to help keep track of the connections.
My method for soldering wires into the pins is to heat up the eage of the tinned pin with the soldering iron while and insert the wire into the pin while the solder is molten, then quickly move the iron away.