How do aircraft windshields not crack when flying at supersonic speeds through precipitation?
The Concorde had 2 discrete windshields for different stages of flight. During subsonic flight (mainly take off and landing) the nose was drooped and the inner windshield was exposed. For supersonic flight the Concorde would raise its drop nose which also contained an outer windshield that would cover the inner windows. This outer windshield was made of special heat resistant glass that was also tinted. It can be assumed that this outer windshield could also withstand any pressure exerted during supersonic flight. The bigger problem in the Concorde was heat: the nose had to have fuel pumped through it just to keep it cool enough. For what it's worth the Concorde flew at about 50,000ft when it was supersonic; this is well above most weather systems.

Drooped

 
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In Place 


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If you are interested, there is an old paper you can find here that covers windshield failures and lists the Concorde as the #2 aircraft for it (747 comes in at #1) over the years the paper covers. Interestingly enough here is a table of causes:  



Heat fatigue and manufacturing defects (what I assume they mean by saying the windshield itself is the issue) are the primary cause of the issue according to this data. Here is one of the patents filed for what appears to be the Concorde windshield; it's an interesting read!

Going further, if we look at the most extreme case of supersonic planes, the SR-71 Blackbird, the inside of the windshield was in excess of 120&deg;C and the exterior of the nose area would be closer to 260&deg;C during supersonic flight. This would turn any water into vapor if it came into contact with the surface. While I don't know the specifics of that, the Blackbird flew at close to 80,000ft, so it too was above most (if not all) weather. I can't yet find an official quote of what that windshield was made of, but heat was still a larger issue than hitting rain. 


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Here is an interesting paper on what happens to water droplets when they suddenly encounter a high-speed airstream. It might suggest that such an impact with an aircraft would also cause vaporization, aside from the heat of the surface causing it. 

Even subsonic aircraft windshields are extremely strong; they can take continuous hits from solid hail chunks and not shatter. 


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