How can I extend silicon hotbed wires safely?
750 W at 120 V is 6.3 A. 22-24 AWG is on the thin side for this. I would recommend 18 AWG or thicker. You don't need a specific style of insulation for this (other than something that is rated for the voltage and temperature the wire will need to withstand, but most commonly found wire should be good).

A good way of connecting the wires would be to solder them. If you do not want to solder, there are many products on the market for connecting wires. A butt connector that you crimp could be a good option, or you could use a WAGO clamp. Whatever option you end up using, be sure to provide adequate strain relief as the connection point (be it soldered or with a connector) is more likely to fail from fatigue.