I'm trying to replicate the visual effect of this product box:
How is this created? What type of materials should I use?
This is a simple application of the moiré effect. Here's one way to recreate it:
Ps. For this quick online demo, I used GIMP, but for an actual printed product you'll almost certainly want to use a vector graphics editor, which will give you more precise control over the mask width and placement.
You can also experiment with variations of this technique, such as varying the stripe / gap widths, or with shifting the text mask slightly for each letter in the text instead of just scaling it, which should cause each letter to appear at a different time (and may well be what's actually done in the examples you show above).
Of course, the tricky part, if you want to make this into an actual physical product, will be getting the card and screen properly aligned, so that the text will be initially hidden.
Funny that you ask this - I just saw this for the first time last week myself in book form at a toy store. I searched around for that online and found this:
Not sure if Scanimation is just a brand name or not, but here's their description:
The video gives some very brief insight into the process. If someone here has experience with this sort of thing they'll be able to give you a better answer, but hopefully this will point you in the right direction until then.
EDIT: Props to @Ilan for his comment, where he references a tutorial on how to recreate the effect.
P.S. The page I link to above also says "It harkens back to the old magical days of the kinetoscope" - that might be worth looking into as well. People had some innovative ways of animating things before traditional film happened.