Indeed, the word is often used in sports and politics: when one team really loses badly, the losing team has received a "Klatsche" – or the wining team gave the losing team a "Klatsche".
But the word has also a broader meaning - and I suppose this is the origin of the above usage: ironically "klatschen" is actually the german word for clapping. But the meaning of this clapping is not only applauding – but also something like slapping. For example you kill flies with a "Fliegenklatsche" or short "Klatsche" - a "fly slapper" / "fly swatter".
And then there is the very bad street slang – if you're planning to beat somebody XY up you could ask your gang friend to joins in: "Lass uns XY klatschen" (="Let's clap XY) which basicalley means "let's beat XY up".
Not that bad is yet annother meaning, which is close to "a slap in the face" –– so if somebody asks you "Soll ich Dir eine klatschen?" you better say no – because literally it would mean "shall I clap you one?" which stands for "would you like to get a slap in the face?".
For "gossip" a shorter version is used: "Klatsch" – and it is actually male "der Klatsch" – but the term you're asking about is female "die Klatsche" – but I guess they are still somehow related. When people meet for coffee and gossipping they also sometimes call it "Kaffeeklatsch".
So I suppose the sports and politics related meaning of "Klatsche" comes from somewhere between all those meanings mentioned above – and I personally would describe it as a team feeling of getting a slap in the team face.
See also http://www.dict.cc/?s=slapping