Why is it that self-professed "nerds" are always so blatantly anti-intellectual?
Anti-intellectual is the wrong term. Anti-system or anti-academia would be better. The nerds you refer to generally believe strongly in learning and becoming smarter (indeed, knowledge and intelligence are measures by which many nerds brag) but not so strongly in the system of academia, "qualifications" and systems of education.
I believe the argument is that claiming that a weekend hacking Perl is of greater value than an entire year's worth of academic school work undervalues skills like reading and writing, history, even math and the sciences. Maybe if you are arguing that you knew the entire curriculum from front to back before entering the 11th grade then one weekend of Perl might be of greater value. But if there are even a few things in each subject you didn't know before hand then claiming that a weekend of Perl is more valuable is just hyperbole.
Anti-intellectual is the wrong term. Anti-system or anti-academia would be better. The nerds you refer to generally believe strongly in learning and becoming smarter (indeed, knowledge and intelligence are measures by which many nerds brag) but not so strongly in the system of academia, "qualifications" and systems of education.