Hi, Shelant:
I you are trying to do a design using pre-programmed stitches with a regular sewing machine or an embroidery pattern using an embroidery design with a hoop, you need to stabilize the fleece. There are many types of stabilizer and stabilizing techniques, so all I can offer is some very basic info.
Fabric needs to have enough strength to hold itself together while it's beeing pummeled with needle insertions and wads of thread. That's where stabilizer comes in. It supports the fabric so it can be embroidered – stabilizing it.
If you are using an embroidery machine or embroidery attachment, you would hoop a piece of stabilizer and then adhere your fleece to the hooped stablizer before starting the embroidery. The design size and stitch density would determine the type(s) of stabilizer to use.
If you are doing some free-motion machine embroidery or using those pre-programmed stitches/designs, you still need to stablize you fabric. Your fleece weight, design, stitch density determines the type of stablizer to use. Your fleece is snagging because there is no support under the fabric.
Stabilizer is available at most fabric stores and many places online. Without a better description of your design, I'd suggest you take a piece of medium weight stabilizer and adhere it to the back of your fleece with a temporary spray adhesive and then try your design. If your thread/fabric continues to get caught in your machine, then the stitch density might be too dense for the fabric/stabilizer combo. You might loosen the design, adjust thread tensions, etc. to see if that helps. You can also search the web for articles on how to embroider fleece. Embroidery Machine Essentials (Fleece Techniques) by Nancy Cornwell could be another helpful source.
Hope this helps! 