The link posted before,
https://twitter.com/brainsmoke/status/948561799875502080 , is currently the only public demonstration I know, but you can see how it works in general --- if an address has been recently accessed, then it will be in the cache so it will be faster to access than one which hasn't. My guess is that the CPU will do a speculative access and cache the data even if the access turns out to be invalid, altering the timing thereafter.
Intel's response that it's "operating as designed" is because no one ever thought this would be a real problem, and so far it remains to be seen how much of one it really is.
Is this even an issue for standalone PCs ?
Yes - your applications aren't meant to be able to find the kernel, let alone read it.
It depends on what applications you run, and whether you trust them. Obviously if you trust everything running on the CPU, e.g. like in an embedded system, this has little relevance. If you're a cloud provider or user with hardware being shared by dozens if not more users who don't trust each other at all, then it's a big problem.
This also theoretically includes things like Javascript running in browsers, so you need to be careful of
any untrusted code running on your system, but if you don't have any, the situation hasn't changed.
It will be interesting to see what happens...