Background
In many textbooks on electromagnetism, the sequence in which topics are introduced are generally as follows — electrostatics, magnetostatics, electrodynamics, wave propagation in free space, and wave propagation in confined space, in that order. As an example, consider Griffiths' 3rd edition of Introduction to Electrodynamics. Similarly, consider Orfanidis' Electromagnetic Waves and Antennas.
By using the discussion on wave propagation in free space as a launchpad, the discussion on wave propagation in a confined space usually introduces some additional boundary conditions that the solutions to Maxwell's equations for wave propagation in free space did not have.
If a proposed solution can be plugged into Maxwell's equations and not produce any contradictions, then they are valid. The goal is then to maintain the monochromatic wave solutions that worked for free space wave propagation with some "patches" to account for the fact that the propagating EM energy will be confined by the hollow waveguide. The reason monochromatic (one-wavelength) wave solutions are desirable is that, by superposition, any signal can be expressed as a sum of one or more monochromatic waves, and so any signal's behavior in a hollow waveguide can be understood.
One of these "patches" is the boundary condition that $E_{\parallel}=0$. This is because the net electric field in an ideal conductor, such as the material that makes up the confining hollow waveguide, must be $0$ because any externally applied electric field will influence electrons to rush to counter the externally applied electric field. If there is no net electric field (meaning $\vec{E}=\vec{0}$), then there is no net electric field in the tangential direction ($E_{\parallel}=0$) along and inside the metal.
If there is no net electric field along the metal, then, by Faraday's law, there must be no net changing magnetic field $\frac{\partial \vec{H}}{\partial t}$. This means that each of the components $\frac{\partial H_x}{\partial t}$, $\frac{\partial H_y}{\partial t}$, and $\frac{\partial H_z}{\partial t}$ must all be $0$ too. If there is no initial magnetic field (meaning $\vec{H}=\vec{0}$, then there must be no magnetic field at any later point. Therefore $\vec{H}=\vec{0}$ and $H_{\perp}=0$.
So these are the two additional boundary conditions that must be considered. An implicit assumption is that the material on the interior of the waveguide is isotropic, homogeneous, and linear. This means that $\vec{B}=\mu\vec{H}$ and therefore $\vec{B}=\vec{0}$ and $B_{\perp}=0$ at the boundary of the metal + hollow interior interface as well.
Problem
Frequently, the monochromatic wave solution to Maxwell's equations given the boundary conditions of a hollow waveguide are presented as
$$
\vec{F}\left(x,y,z,t\right)=\left[\text{spatial component}\right]\left[\text{temporal component}\right]
$$
For example from Orfanidis,
$$
\vec{E}\left(x,y,z,t\right)=\vec{E}\left(x,y\right)e^{j\omega t – jkz}
$$
and from Griffiths,
$$
\vec{E}\left(x,y,z,t\right)=\widetilde{\vec{E}}_0\left(x,y\right)e^{i\left(kz – \omega t\right)}
$$
Further, $\vec{E}$ is a 3-dimensional vector with components $E_x\left(x,y\right)$, $E_y\left(x,y\right)$, and $E_z\left(x,y\right)$.
Bottom line: Why is the spatial component only a function of $x$ and $y$? Further, why does spatial component have $x$, $y$, $z$ components [e.g. $E_x$, $E_y$, and $E_z$] (where $E_x$, $E_y$, and $E_z$ are each functions of only $x$ and $y$)?
Best Answer
To begin with, Maxwell's equations are vector partial differential equations, which makes their solution intrinsically more difficult than that of scalar partial differential equations.
To help obviate this difficulty, an ansatz was long ago made (by Hertz himself, perhaps?) who proposed two useful categories of wave solution: transverse electric (TE), and transverse magnetic (TM). The term 'Transverse' here means 'transverse to the waveguide axis', which is conventionally the 'z' direction, and the presumed direction of propagation.
These categories are somewhat self-explanatory. TM waves have no z directed magnetic field component, and TE waves have no z directed electric field component. Each of these is derivable as the curl of an appropriate vector potential, magnetic or electric. If the idea of an electric vector potential is troublesome to you, try the book by Harrington 'Time Harmonic Electromagnetic Fields.' (With no denigration of Griffiths, I advise that you widen the scope of your reading.) In either case the vector potential is assumed to have a single, z-directed, component, comprising a scalar function of the form f(x,y)exp(ikz). This entire scalar function is presumed to satisfy the scalar Helmholtz equation (with propagation constant k), which insures that taking its curl (and the curl of the resultant) will lead to Maxwell's equations.
In essence, TE and TM waves are assumed; these ideas came out of someone's brain. Their justification is that they lead us (as noted) to Maxwell's equations, and from there to innumerable useful solutions and applications. Inasmuch as a waveguide is designed to propagate a monochromatic wave of some chosen frequency, the simplest form of traveling wave solution is exp(ikz), as noted. As far as why one wants a monochromatic wave, one answer would be that (at a guess) >90 % of communication electronics involves the modulation of a carrier wave, with the frequency bandwidth of modulation typically small in comparison to the carrier frequency.
Sorry to go on at such length. I hope this helps.