(Adding to the existing answers)
Light must move at $c$ (locally) in every inertial reference frame. In a reference frame belonging to light, this rule would break, so such a reference frame does not exist / is not valid for consideration.
Consider that to transmit light from point A to point B, it has to cross some physical distance. This 'crossing' takes time from the point of view of any inertial frame of reference (or someone who is standing still).
However to light, this 'crossing' is instantaneous. It is not immediately clear why this should result in any oscillations... To assist with this, an image from this article:
Where $\vec B$ is the magnetic field, $\vec E$ is the electric field and $\vec v$ is the direction of "travel".
Without going neck deep in math, the shapes of the red and blue curves are sinusoidal. Notice how they are exactly aligned. Other 'alignments' (polarizations) are possible but this is the simplest to picture.
What you need to understand is the $x$ axis here is actually starting and ending in two different times - $x$ is both a distance and a duration (when using suitable units in terms of $c$). What this diagram shows is how the electric and magnetic fields change over time and space during the propagation of light.
The fields increase / decrease in strength in a perpendicular direction of travel, due to the other field changing. This self-influence is what travels at light speed and makes up light (rather the energy carried by this self-influence). But in order for this self-influence to be possible, the fields need to be changing in strength in a particular way through space and time.
This gives rise to oscillations for observers that can't move with light / at light speed.