Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

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.

One answer is that property of being complex analytic is equivalent to being a solution to a differential equation (namely the Cauchy-Riemann equations) whereas there is no analagous formulation for being a smooth function. Once you have this formulation it should be immediately clear that complex analytic functions are rigid because solutions to differential equations are rigid. The whole idea of boundary value problems and initial value problems is predicated on the fact that knowing a solution to differential equation in a small area determines its values everywhere, which is precisely the type of rigidity that complex analytic functions have.

So you might ask why do complex analytic functions satisfy the Cauchy-Riemann equations. Well every real smooth function from R^2 to R^2 has a derivative, which is a 2x2 matrix. Requiring the function to be complex differentiable is the same as requiring that matrix to be a "complex number", i.e. a matrix of the following form: first row = [a, -b], second row = [b, a]. Well this condition is precisely the Cauchy-Riemann equations.

The book Visual Complex Analysis gives a good explanation: locally, analytic functions are rotations and dilations. Disks go to disks. A smooth function of two real variables may map disks to ellipses. That is, a real valued function can distort disks in a way that analytic functions cannot.