As I mentioned, Bad Königshofen is sort of representitive for an average sports club in Germany.
Clubs itself, and the culture associated with it, are deeply rooted within the german society. Whether they are sports clubs, stamp collector clubs, oldtimer clubs, local history clubs, old train clubs, Schützenvereine (
https://en.wikipedia.org/wiki/Schützenverein; http://www.schuetzenverein-niederaden.de/ ) etc.
In the US you have High School and Varsity sports (where you compete against each other and these schools oftentimes have their own long lasting traditions), that is basically non-existent in Germany. You do have psycical education at school and many colleges/universities have at least some sort of a sports club, but they are not really competing against each other; the college sports clubs are predominantly purely recreational.
If you want to do sports somewhat competetively, you need to become a member of a sports club.
Many of these clubs were founded in the 1800s (Bad Königshofen in 1861, same year as the main sports club in the town I grew up in; the oldest sports club In Germany was founded in 1814). Of course back then the "portfolio" of the available types of sports was very limited. Mostly only gymnastics and/or physical education was offered. Later other sports were added, such as soccer, volleyball, handball, tennis, weigthlifting, badminton, chess etc.
These clubs you can find all over Germany. There are some one-sport-clubs, but most of them are multisport-clubs. While you can find more clubs in bigger cities,
sports clubs in particular, as well as "Schützenvereine" and voluntary fire departments play a key and vital role in peoples's every day life in smaller towns and more rural regions. If your parents are members of the local sports club, then you are very likely to become a member of that club as well. Many of these clubs have youth/junior departments and they pay attention to coaching the kids. But aside from all the sports, these clubs serve a social purpose, pretty much the way it is described here:
its a club culture
community gets together to support the local club.
players could be bought in, but the fans are still mostly local.
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The way the club functions, involves family a lot.
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Over time, it becomes part of tradition/life
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it's more than sports basically, its social too
I recall that video I saw was after a league night, all the players, fans stayed on for finger snacks at the bar
Oftentimes you would just meet up at the club house, play cards, drink and barbacue together, watching tv and so on and so forth (mainly so forth...

).
In other words: You simply grow up with it from an early age on and there is usually are very strong identification with the club, which also results in the atmosphere OP witnessed and admired(?) while watching Bad Königshofen vs. Düsseldorf. The members are proud of their club and the players need to be cheered on because they represent the club and the region; needless to say, most of the players are also role models for the kids in the club.
What also helps tt being a fairly popular sport in Germany is what is mentioned here:
More Opportunity to play -> more recreational players -> more fans / audience
You will find stone tables in parks, school yards, lidos, playgrounds. Also, most youth clubs/centres have tt tables. The one in my home town had two and they were quite popular; even some younger adults in their 20s would sneak in in order to play. My old "middle school" had like 7 or 8 pretty usable tt tables in the auditorium and we would play tt during almost every school break.
Table tennis being kind of "omnipresent", because it is readily available in nearby places, and the strong club culture which is deeply rooted within the german society is imo the main "secret"
NextLevel was asking about.