If I had to give a one sentence answer to this question, it would be as follows:
*That the phase and amplitude alone on one plane is enough to wholly define a three-dimensional light field arises from the various uniqueness theorems for Maxwell's equations within a connected volume given the solution on the volume's boundary; otherwise put: once you know a solution on a boundary, then the values within must follow from "reasonable" physical assumptions.
For simplicity let's sit with the scalar diffraction theory, so now we are essentially talking about uniqueness theorems for the Helmholtz equation $(\nabla^2 + k^2) \psi = 0$.
Uniqueness theorems when $k^2 > 0$ or when $k^2 \in \mathbb{C}-\mathbb{R}$ are much more complicated than when $k^2\leq0$. The latter case corresponds to static solutions of the Klein Gordon equation or to static solutions of the Maxwell equations with or without an assumption of a massive photon; see My answer here for more details. Such cases have very strong uniqueness theorems: once a solution's values are set on a compact volume's boundary, there is only one possible solution within the volume. This situation even extends to semi-infinite volumes. However the former situation includes $k^2>0$, the case for scalar diffraction in freespace or a lossless dielectric: uniqueness theorems need further strong assumptions about the field to make them work. Thankfully, some of these assumptions are reasonable physically.
We can restore simplicity to the solutions of the freespace Helmholtz equation (i.e. to the situation we have with a hologram) by making reasonable physical assumptions such as the Sommerfeld radiation condition or that the field is a tempered distribution; for more information on the latter condition, see my answers here or here.
Given these assumptions, together with the assumption that the field is propagating purely left-to-right, we can reconstruct a field from the hologram as follows. You begin with the Helmholtz equation in a homogeneous medium $(\nabla^2 + k^2)\psi = 0$. If the field comprises only plane waves in the positive $z$ direction then we can represent the diffraction of any scalar field on any transverse (of the form $z=c$) plane by:
$$\begin{array}{lcl}\psi(x,y,z) &=& \frac{1}{2\pi}\int_{\mathbb{R}^2} \left[\exp\left(i \left(k_x x + k_y y\right)\right) \exp\left(i \left(\sqrt{k^2 - k_x^2-k_y^2}-k\right) z\right)\,\Psi(k_x,\,k_y)\right]{\rm d} k_x {\rm d} k_y\\
\Psi(k_x,\,k_y)&=&\frac{1}{2\pi}\int_{\mathbb{R}^2} \exp\left(-i \left(k_x u + k_y v\right)\right)\,\psi(u,\,v,\,0)\,{\rm d} u\, {\rm d} v\end{array}$$
To understand this, let's put carefully into words the algorithmic steps encoded in these two equations:
- Take the Fourier transform of the scalar field over a transverse plane to express it as a superposition of scalar plane waves $\psi_{k_x,k_y}(x,y,0) = \exp\left(i \left(k_x x + k_y y\right)\right)$ with superposition weights $\Psi(k_x,k_y)$;
- Note that plane waves propagating in the $+z$ direction fulfilling the Helmholtz equation vary as $\psi_{k_x,k_y}(x,y,z) = \exp\left(i \left(k_x x + k_y y\right)\right) \exp\left(i \left(\sqrt{k^2 - k_x^2-k_y^2}-k\right) z\right)$;
- Propagate each such plane wave from the $z=0$ plane to the general $z$ plane using the plane wave solution noted in step 2;
- Inverse Fourier transform the propagated waves to reassemble the field at the general $z$ plane.
If you can understand these steps you should be other see how the solution to Helmholtz's equation, i.e. the full three-dimensional scalar light field, is reconstructed from its values on a plane. The latter of course is what a phase and intensity mask hologram encodes.
What hinders holography? I am not up with the latest hologram production techniques, but essentially making a hologram is a kind of interferometry and as such calls for low vibration and building of an interferogram between transmitted and reference light. One can't simply "snap" a hologram like one can with a digital camera (or even with an older style film camera). Moreover, the phase masking needed to make the equations above work is highly colour-dependent, so that any kind of colour holography is even more restrictive than the making of one-colour holograms. The Holography Wiki page gives you a good overview; the "rainbow" holographic technique is the nearest I know of to colour holography. Aside from this technique, most holograms need high coherence in the light source for reconstruction.
Another interesting technique is the manipulation of light by computer generated holography, where one computes by solving Maxwell's equation the phase and amplitude mask needed for e.g. nulling out the mean aberration from a lens before analysis by an interferometer.
