The conductive ring goes to a shield. There should be lots of vias from the ring to an inner-layer ground plane. If the vias are not visible, they might be implemented as filled vias, and you can see them in an X-ray view of the board. The number of vias required is not easy to generalize, but I use a minimum of 10 per inch. These vias should have no thermal relief. In general, any two conductors on an inner layer that traverse from outside the ring to inside the ring should have two vias separating them.
In my circuits, the purpose of the rings is for mounting shields, as shown in the answer by Some Hardware Guy. Most often, the purpose of the shields is to provide RF isolation between the circuits. For example, shields keep the output of an RF filter or amplifier isolated from the input. Also, the shields help prevent external interference from switching power supplies, digital circuits, etc. Less often, shields prevent leakage of RF signals to the outside environment, for example to meet FCC interference requirements.
These shields don't do much unless there is some sort of gasket or other way to get a very low resistance contact all along the shield and the board. Just screwing the shield down really tight is insufficient. Seams are leaky. Since solder-mask is an insulator, it is kept off of the ring to allow for a good electrical contact.
The ring needs to be wide enough to allow room for the footprint of the shield and gasket.
The TI layout linked in your question looks like an RF circuit block that was copied into the design. It doesn't look like it is set up to mate with a shield. Here is a sketch from a brief article about designing radios into products: