What are the mechanisms used to avoid destruction of space probes and satellites during meteor showers? Are they affected by them?
Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space, listen...
Okay, obligatory Hitchhiker's Guide quote out of the way, that's the essence of why they're not destroyed. Space is big, satellites are small, and a meteor shower is incredibly sparse. Even an incredibly dense shower like the Leonids (~1000 meteors/hour) is still only around one meteor per 15,000 square kilometers per minute; a typical satellite has only a 0.0000001% chance of being hit during any given one-minute period.
There's also the size of the meteors to consider. That bright streak you just saw flash across the sky? It's the size of a grain of sand. Yes, it's moving fast enough to cause damage, but it won't "destroy" a satellite, it'll put a small crater in whatever it hits. For example, it might destroy a single cell in a solar panel: annoying, but hardly fatal.
Some precautions are taken (for example, during the Perseids, the Hubble Space Telescope isn't looking anywhere near Perseus), but mostly it's a matter of playing the odds.