As a counterpoint to gowers's devil's advocacy, I'd mention that some formulas for $\pi$ have been discovered experimentally, and in some cases we still don't know how to prove them. For example, in the paper "About a New Kind of Ramanujan-type Series" by Jesús Guillera (Experimental Mathematics 12 (2003), 507–510), the following conjecture by Gourevitch is stated: $$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$ As far as I know, this is still unproved. In principle this kind of formula should be WZ-able, but it seems to be just out of reach of current computers. And probably there ultimately does exist some "motivic explanation" as Torsten Ekedahl said, but since we don't currently know of one, I think that one answer to Qiaochu's question is, "experimental observation."