A plane electromagnetic wave may be understood as the classical limit of a "coherent state" of photons, the latter of which is made up of an indefinite number of photons. In particular, the number of photons contained in a coherent state obeys some probability distribution, but in the classical limit that distribution is peaked at some very large number with very little spread around it, so one can characterize the state by N, the average number of photons in the state.
Just like with the number of photons, the electric and magnetic fields are also uncertain, taking values from some probability distribution. However, if you take the expectation value of the electric and magnetic fields, you find that they are also very sharply peak about some mean values, and that mean value evolves in space and time just as in the picture you've posted. This evolution can be represented mathematically as
$$
<|\mathbf{E}(x,t)|> = \sqrt{\frac{N \hbar}{\omega \kappa}} \cos(\omega t - kx)
$$
where \omega is the frequency of the photon, k = \omega/c, \hbar is the reduced Planck's constant, and \kappa stands for some mess of constants (depending on your choice of unit system) which I will not bother to figure out. I will also point out that the average energy of this state is given by the usual Planck formula formula
$$
<E> = N \hbar \omega.
$$
We now take the classical limit, which clearly requires \hbar \rightarrow 0. However, doing so naively results in the amplitude and energy of the wave going to zero. Instead, we must take the limit
$$
\hbar \rightarrow 0, \quad N \rightarrow \infty, \quad N \hbar \rightarrow \mathrm{constant}.
$$
The constant can be any real number. Therefore, in the classical limit we take the number of photons in the wave to be infinite. (This is as close as your question comes to having an answer.) With such a limit, we also see that energy is proportional to the square of the amplitude of the electric field. The constant which appeared is a tunable parameter in the classical theory (it just corresponds to boundary conditions when you solve Maxwell's equations).
Finally, let me stress that a single photon of a given frequency looks nothing like anything you've seen in classical physics. In particular, the electric and magnetic fields of a single photon are constant in time, and they are (as always in quantum electrodynamics) described by a probability distribution. The average fields of a photon are actually zero, but there is some finite spread in probabilities around zero. This is not something that looks like classical electromagnetism.
