It exploits the fact the regexp will always fail. Note that the (?{$=++}) always succeeds, adding 1 to $= each time it's executed. The trailing ^ cause the pattern to fail. The (^)(1|11\1)* will always succeed, but in such a way the optimizer doesn't spoil things.

Now, because (1|11\1) eats either 1 or 2 characters, and then tries the trick on the rest of the pattern, the number of ways a string of length n can fail is:

    F(n) = F(n-1) + F(n-2)
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which is the formula for Fibonacci numbers.

Also note that $==1 just means $= = 1.