What is the purpose of lantern ring and packing box ring in the sealing picture below?
The packing box ring provides a square, flat surface that is perpendicular to the bore for the packing to be pressed against. It has a tight fit against the packing bore and shaft so the packing does not extrude into any gaps at the OD and ID. If the packing pressed directly against the as machined bottom of the packing bore, it can create uneven loading by the angled surface, plus there is a larger gap between the body and stem for the packing to extrude into.
Lantern rings serve many different purposes. Grease injection is one, but is not very common as lubricant needs to be fully compatible with the media in the valve. Most common is for a leak-off port that is machined through the side of the valve body and into the area of the lantern ring. This port can be used to check if the lower set of packing is leaking or not.
Another common way this type of arrangement is used is to inject pressure. It is common to inject gas or liquid pressure into the port that is slightly higher than the process pressure inside the valve. This helps to ensure that whatever is inside the valve cannot leak externally.
Above is a snippet.