The PCB stackup for stripline must be GND, dielectric, signal, dielectric, GND; three metal layers separated by two layers of dielectric material. The Wikipedia description is telling you to make sure that the two GND planes are both at the same voltage potential, so you'd want to connect them together electrically. This is done by running vias to tie them together. Generally, this is done several trace-widths away from the edge of the signal conductor. Think of it like two rows of fence (vias), running parallel to a sidewalk/path (stripline center conductor).
(Top view, looking down, "through" the PCB)
vias o o o o o o o o o o o o o o o o o o
(gap)
stripline ////////////////////////////////////////////////
(gap)
vias o o o o o o o o o o o o o o o o o o
You'll want to use a lot of vias so that you get the benefit of lower resistance/impedance due to multiple parallel conductors (vias). Spacing < lamda,min/10
The article says nothing about shorting a power plane and ground plane together. As a side note, DO NOT use a power plane for one of your ground planes in a stripline design (that should be obvious from the names).
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And in a DDR2 routing example as shown in the attached picture, I noticed a ribbon-like band of vias connected to GND, which are in turn connected by microstrip (NOT stripline) tracks surrounding the DDR2. They are more clearly shown in the image than the verbal description here.
What is the purpose of this GND via band? Does it have anything to do with the "shorting two ground planes" concept and the "a row of vias running parallel to the strip on each side" treatment?
The ring around the outside of the board is a "guard band", used to isolate the signal spaces (inside the DDR chip, and outside).
Any electric fields generated on the DDR chip, trying to go outside of the circuit, will prefer to terminate on the ground ring. This keeps the loop-path short, and keeps them from radiating. Also, any signals generated outside the DDR chip will terminate on the ground ring, and return to their source via the common grounding of the entire assembly. This keeps outside E-fields from terminating on the DDR signal lines, inducing unwanted signals onto them.