We say the electromagnetic wave is oscillating because something waves as the wave passes by. Light does propagate as per the above image, but it isn't the full story. For a bit more, have a look at the Wikipedia electromagnetic radiation article and note this:
"Also, E and B far-fields in free space, which as wave solutions depend primarily on these two Maxwell equations, are in-phase with each other. This is guaranteed since the generic wave solution is first order in both space and time, and the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time".
Also see the Aharonov-Bohm article about potential being "more fundamental" than field, and think along these lines: the sinusoidal electric waveform is the spatial derivative of electromagnetic four-potential, whilst the sinusoidal magnetic waveform is the time-derivative. To picture this, imagine you're in a canoe at sea. Imagine an oceanic swell wave comes at you. Imagine it's just a hump of water, without a trough. As the wave approaches, your canoe tilts upward. The degree of tilt denotes E, whilst the rate of change of tilt denotes B. When you're momentarily at the top of the wave, your canoe is horizontal and has momentarily stopped tilting, so E and B are zero. Then as you go down the other side, the situation is reversed. In similar vein you can draw the sinusoidal electric waveform as the spatial derivative of the "hump" of potential like this:
Now, note this: if you flew over the oceanic swell wave in a helicopter, keeping pace with it, it's just a hump. It isn't oscillating at all. It's only oscillating for the guy in the canoe, because it's passing him by. It's similar for the photon, which I think of as one electromagnetic wave. And of course waves usually come in "wave trains", wherein one wave follows another, as per your picture, because the wave was generated by something oscillating.