Once upon a time there was a show on TV called "The Secret Life of Machines". The second episode of the show was "The Secret Life of the Sewing Machine".
Here is a short 5-minute version of the show, just showing the part about how the sewing machine makes a stitch. Everyone who wants to understand what is really happening when the sewing machine makes a stitch should watch this video.
The whole show is 25 minutes long. YouTube has a 15-minute limit. Below is the whole show, split in to two parts.