What's the correct way to safely splice into a wire meant for mains voltage
The best way to do this type of wiring would be using crimp connections such as spade lugs or quick-connect lugs, making sure everything uses the male counterpart. 

They come in various shapes and sizes. For mains, I would suggest getting the shielded type. (Picture shows unshielded version so it's clearer what type I'm talking about)  

You need the right type of tool to crimp these in.