Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
Well, real-valued analytic functions are just as rigid as their complex-valued counterparts.  The true question is why complex smooth (or complex differentiable) functions are automatically complex analytic, whilst real smooth (or real differentiable) functions need not be real analytic.

As Qiaochu says, one answer is elliptic regularity: complex differentiable functions obey a non-trivial equation (the Cauchy-Riemann equation) which implies a integral representation (the Cauchy integral formula) which then implies analyticity (Taylor expansion of the Cauchy kernel); the ellipticity of the Cauchy-Riemann equation is what gives the analyticity of its fundamental solution, the Cauchy kernel.  Real differentiable functions obey no such equation.

Another approach is via Cauchy's theorem.  In both the real and complex setting, differentiability implies that the integral over a closed (or more precisely, exact) contour is zero.  But in the real case this conclusion has trivial content because all closed contours are degenerate in one (topological) dimension.  In the complex case we have non-trivial closed contours, and this makes all the difference.

EDIT: Actually, the above two answers are basically equivalent; the latter is basically the integral form of the former (Morera's theorem).  Also, to be truly nitpicky, "differentiable" should be "continuously differentiable" in the above discussion.
