Early applications of $e^{i\omega t}$ in the context of electromagnetism were understood as a mathematical device: the physical fields are real, and the complex exponential is a convenient method to implement trigonometric relations. The earliest application I know of where $e^{i\omega t}$ acquires a physical significance is by Erwin Schrödinger, who needed it to describe the time dependence of the electron wave function. He initially believed, or hoped, that this would eventually also turn out to be mathematical device. In a 1926 letter to Hendrik Lorentz he wrote:
"What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. $\psi$ is surely fundamentally a real function."
It did not work out that way, the $e^{i\omega t}$ factor is still with us, and indeed just this week we learned of a Experimental refutation of real-valued quantum mechanics under strict locality conditions. For more on this, see Schrödinger’s bewilderment – Quantum theory needs complex numbers.
One might wonder, why not just work with standing waves of electrons, thereby eliminating the complex phase factor? There is a fundamental difference here between factors $e^{ikx}$ and $e^{i\omega t}$. The former can be reduced to a real sine or cosine by superposition with $e^{-ikx}$. The latter can not, the reason being that superpositions of $e^{i\omega t}$ and $e^{-i\omega t}$ are forbidden for electrons (such a superposition would couple a particle to antiparticle, violating charge conservation). There may exist charge-neutral particles which are their own antiparticle (socalled Majorana fermions), and indeed, in that case a real wave equation applies.
Suggestion for further reading: A road to reality with topological superconductors.