Alligator clips are fine for experiments, but if you're building an actual project you'll want something more secure. Here are two methods of connecting power to the conductive fabric: rivets and snaps.

The rivet option is very secure, while the snap option allows for rotating or removing the wires after installation. Both methods delivered current to the fabric reasonably well, but results were varied, and you'll need to do some tests to choose a method that works for your specific project.

These techniques are based on the research and methods documented by Kobakant at their website, How To Get What You Want. Check out their Hard/Soft Connections for more ideas and techniques!

Snaps

To use snaps, solder your wire to the back of the stud (the "innie" side of the snap). Then punch a small hole in the heater fabric and thread the post (the "outie" side of the snap) through the hole. Snap the post and the stud together, sandwiching the heater fabric.

Eyelets

For conductivity, make sure you're using bare metal eyelets, do not use eyelets that are coated or painted. Start by bending the stripped end of the wire into a hook shape. Small jewelry pliers can be helpful here.

Press the hook around the post of the eyelet. Punch a hole in the conductive fabric and feed the eyelet into the hole. Crimp with eyelet pliers to secure, catching the wire inside the rim of the eyelet.