Here is a PDF of the Instructable that includes all the secondary images and annotations.
Here is a video that shows how it works.
Another video of assembly to help if you are having trouble working out some of the steps.
If you want to build one and don't care as much about the appearance you can try the quicker and easier method.
Step 1: Things You'll Need
Small rubber bands
Something heavy and flat (like a large book)
Cardapult design (attached in following steps)
Other stuff that may come in handy:
Something heavy and hard (like a hammer)
For the paper I had a hard time finding anything thick enough locally except for the perforated sheets for DIY business cards. These will work, but alignment is more difficult since the edge of the card is already made. You'll need something about .012" or thicker. Which corresponds to 90 lb Cover and 140 lb Index (card stock). If you are just making these for fun then old folders or thin cardboard product packaging will work great as well.
I tried a few different glues and the best I found for these purposes was an Elmer's Craft Bond Glue Pen
Step 2: Print Card Text
Go to Tools>Options>Print and make sure "Drawing Objects" is unchecked.
Or get to the same screen by hitting Options at the bottom of the File>Print dialog
Or just delete the background image after you have things lined up the way you like
Free download link on next step includes this template
Step 3: Cut Card Parts
If you have a similar cutting machine that uses the Robo Master software then you can just grab the attached .gsd files.
If you have a laser cutter or other kind of cutting machine or just want to tweak the design then I'm also attaching the Inkscape .svg files. Inkscape has a lot of output options, so you should be able to get it in whatever format you need from there.
If you are cutting by hand then get out your scissors and hobby knife and figure out the method that is best for you. Here are a few approaches you might consider:
Print designs on sheets and cut along the lines one at a time.
Print it out once and make template pieces that you can either trace or use as a cutting guide.
Glue sections of paper together before cutting them with either of the two methods above
Note that I've purposely made the holes and axles 1/4" so they can be cut with a standard hole punch.
There are three different files
Insides = full sheet of the inside components
Outsides = full sheet of the outside components
Combined = 6 inside components and 4 outside components. Enough to make two cards if you are cutting the parts out individually.
Since Instructables now requires a Pro account to download files, I've uploaded them to my site so you can still download them for free. Download here
Step 4: Glue Inside Parts Together
Also glue together 3 layers of scrap material that you can cut later use for the axles, don't bother trying the glue the actual little circles you cut out together.
Let the glue dry enough that you are sure they won't slide around anymore.
My gluing technique was to pour out a small puddle of the glue on a piece of scrap material and then use a toothpick to scoop up the glue and spread it evenly over the surface of the part. After you get the layers together then put them under a large book to make sure they dry flat.
Step 5: Glue Insides to Back Face
The large L piece lines up in the corner and flush against the notch on the other side.
The skinny piece lines up against the bottom edge of the card. Personally I've started trimming about 1/16" off of a short edge so you don't have to be as exact with the side to side alignment.
The odd shape piece goes on the smaller part of the outside face with the long edge aligned against the edge of the card.
Again give the glue some time to dry before proceeding.
Step 6: Glue on Axles
Place the arm in the small piece you assembled last step and trace the hole location. Carefully glue the circle over your trace marks and carefully do some double checking that the arm will still line up correctly and also make sure you're not gluing the arm to anything.
Repeat with the lower hole. You'll need the upper component in place to make sure the lateral alignment is correct, but vertically the arm should be against the edge of the large L piece.
Again letting the glue dry some before proceeding is recommended.
Step 7: Glue Front Face
Glue needs to be applied to every inside component that is glued to the back face. The same things that were glued in steps 5 and 6 are glued all at once in this step. The legs can be slid into place after letting this step dry for a little bit.