What exactly is the story of this game and how does it connect to the other Wolfenstein games?

Im not sure if there is a story exactly..its more of missions that you play out to blow something up or move a tank etc etc...so each map is its own story that gets played out by both the teams. Its PvP and not like the story mode of single player Wolfenstein.
My best guess at your question

Are you asking how WWII fits into the Wolfenstein lore?

Are you asking how WWII fits into the Wolfenstein lore?
No, I'm asking how the stories of this game do

There isn't really a cohesive story. There are a handful of scenarios that are vaguely WWII like. It wasn't really until Quake Wars where they were really trying to build something resembling a world. Brink was a stepping stone for Dirty Bomb, which I feel is a good title, despite its abandonment.

The multiplayer doesn't have a story, it used to be bundled with Return to Castle Wolfenstein which contains most of the story, in that game you were some sort of spy trying to kill hitler

What exactly is the story of this game and how does it connect to the other Wolfenstein games?
It never used to fit into the classic Wolfenstein story genre, it was more a case of 'extra multiplayer missions' than anything else. However, this has all changed with the Wolfenstein Community creating their own singleplayer version of Wolfenstein: Enemy Territory.
Here's a link:
https://www.moddb.com/mods/et
Here's the video trailer:
▶Watch on YouTube

The two official campaigns full of 3 maps in the game based in Europe and North Africa are based on the RTCW general landscape and the RTCW console exclusive North Africa mission respectively.
By the way did this game kinda create payload in tf2/ow? The Goldrush map sure ends up like a payload lmao