It isn't rocket science and I'll be up front in saying that this project is probably more about using the right(purpose made) materials than any clever technique.
Step 1: The stuff
You can order those online or find them at most any scrap-booking store.
The PVA glue is designed to bond paper like no other glue I have ever encountered and book binders cloth is designed to take a lot of abuse from handling and bending as well as having a paper backing that lends itself to working with PVA.
You'll also need a broken book to repair. I'll leave that up to you.
Step 2: Measure and cut
You'll also want to leave just a little extra to fold over for a more finished look. Little ones won't much care about this "now" but maybe if their favorite book survives into their adulthood they'll be touched by the loving care that went into saving their cherished reading material.
So, with the book closed wrap a tape measure around the edge to see how wide your book cloth will need to be. Then measure the length of the book and add and inch to give yourself a half inch or so to fold over on either end. This ended up being close to 12 x 6 for my project.
I used scissors but you could use anything that will give you a clean line. Be sure to play attention to the grain of the cloth as it is readily noticeable if it's off even a little.
Step 3: Glue in the spine
You'll also want to make a couple small cuts so you have tabs the width of the spine to fold over and glue to the spine itself. This allows for a clean fold and nicer edge on the cover as well as the spine. Easier to see in the images if you just follow those.
Step 4: Repair the binding
I wrapped my cloth fairly snug to help hold the book together. PVA dries very quickly so be as precise as possible when gluing it down. You likely won't get a second chance.