The rope is extremely strong, being made of eight strands of an 11 mm climbing rope, and is based on a military fast rope - developed by Marlow Ropes for getting soldiers out of a helicopter quickly. I stumbled on a video explaining how to braid such a rope on ITS Tactical - it's fairly simple but you definitely need two people to do it when using anything bigger than paracord. Luckily, my wife is a natural at this sort of thing; she watched about 30 seconds of the video and said "got it. Let's go".
Step 1: Materials
A real fast rope costs about $1000. A gym climbing rope costs about $100 (depending on length). 30 m of 11 mm climbing rope is a little over $100. I was lucky enough to get given two retired 11 mm ropes from a climbing buddy - these ropes were past it, having arrested too many falls for safety - but braid eight of them together, and even this old rope is strong enough. In addition to plenty of 10-11 mm diameter rope, you'll need a carabiner, a method of securing the carabiner to a wall, some insulation tape, an assistant or two, patience, an Exacto knife and a BBQ lighter.
Step 2: Prepare for Braiding
You need 4 ropes of equal length for the job - having two different colors will help you keep track while you're braiding, but are (k)not necessary. Attach the carabiner to something at waist height (we used a tie-down attached to my workbench). Thread each of the four ropes through the carabiner so that they double back on themselves. You should now have eight equal sized strands dangling from the carabiner. Tape the top of each rope so that the two strands created by halving it are secured together so that you remember to move the two strands in tandem while braiding.
The carabiner makes it really easy to attach it to a tree and you won't waste length on making an eye splice.
Step 3: Braid
The only difference to the ITS video is that you're doing the braiding on a larger scale. It will take a little while to get the hang of it, because the pairs of rope need to be aligned correctly, but once you've started, it falls in place quite nicely. Assign one person to braid and the other to support the rope and untangle the 8 strands behind the braider. I got given two 60 m ropes, which would have given me a fast rope about 10 m long. However, we don't have anywhere to hang a rope that long, so we just made it out of one 60 m length to make a 5 m climbing rope. The video below should give you a sense of how to do the braid - keep reciting "two under, one over" while remembering to wrap the strands carefully so the braid looks nice and neat.
Step 4: Cut and Fuse
When you run out of rope, tape the end up and cut off with a very sharp blade. Melt the ends of the rope together, and you're done. Our "fast rope" is 40 mm (1.5") thick and 5 m long - not much good for getting out of a helicopter with, but good fun for kids in the backyard...