After grounding the ESD mat and wearing the properly grounded anti-static wrist strap, what is the correct process to take the contents out of the static-shielding bag and take it onto the antistatic work surface (ESD mat) and why?
(The content of the bag I have in mind is a pcb, in specific a single-board computer).
Antistatic precautions mean you should never move a device through a large potential difference.
Once you and the ESD mat are connected to ground, they are at the same 0v potential. Take hold of the bag, this bring it and its contents to 0v. Now you can open the bag and handle the devices with anything that's also at 0v. This includes fingers, or conductive things you are holding. So metal tweezers, 'antistatic' pliers that have special conductive plastic handle covers, cotton gloves (which retain enough moisture to be slightly conductive) but NOT tools with insulating handle covers or plastic gloves.
You may lay the bag on the anti-static mat to place the bag at the same potential as the mat (although you will do this by handing the bag with your wrist strap on). Then, open the bag, remove the contents while wearing your wrist strap, and place the contents on the anti-static mat.
Do not take the board out of the bag without a wrist strap on, and walk it over to the anti-static mat. The act of walking with the board out of the bag can build up static charge. This will then cause a discharge when you place it onto the anti-static mat, possibly causing damage. Wear your properly grounded wrist strap every time you handle the board out of the bag, and only place it on a properly grounded anti-static mat grounded to the same potential as your wrist strap.
