Why do electromagnetic waves oscillate? I've been considering this question, and found many people asking the same (or something similar) online, but none of the answers seemed to address the core point or at least I wasn't able to make sense of them in that regard. I'm looking for a layman-friendly explanation.
I'll copy the clearest exposition of this question I found, from here:

What is it that causes an electromagnetic wave to oscillate? I.e. what is the medium that constrains it and pulls it back from one side to the other?
Discounting pressure waves such as sound, as they push backwards and forwards through air, other waves, with amplitude, need something to constrain them. For instance waves in water are constrained by surface tension and gravity so at the end of the "up" gravity and surface tension pulling the water flat pull it back down and vice versa at the end of the "down".
So what is doing that with visible light, gamma rays, microwaves etc? What pulls their energy back and forth causing them to oscillate?

One of the answers there claimed "The electric field constrains the magnetic field and vice versa." Is this true? I didn't see anyone else making a similar claim.

The above pretty much sums up my question, but for completeness' sake, I'm also including below references to some related threads I found:
This guy asked "Do photons oscillate?". The responses explained nicely how his conception of photons was incorrect, but didn't clarify his question of why the wave forms.
Somebody asked a pretty similar question on reddit: "Why do photons oscillate?", adding "I don't understand what makes them swing back and forth." -- again, the answer explained that photons aren't really "particles of light", and don't oscillate, but again no satisfactory answer to the "why" question was included.
Finally, this page has a great applet demonstrating the basic principle of how disturbances in one charge propagate to nearby ones. The oscillation is provided by a spring for illustration purposes. The text naturally raises the question: "Yeah, that all makes sense, but don't expect me to believe that particles as small as electrons are attached to springs. How is [the first] electron made to wiggle -- I mean, how is its speed or direction of motion changed?" -- but I couldn't find the answer in the following pages.
EDIT: I emphasized some passages above to make it clear I am not assuming the EM field moves in space (I realize what is changing is the amplitude over time), and that what I'm asking is indeed, to quote Bjorn Wesen's answer, "why does the source amplitude oscillate in the first place".
 A: Electromagnetic waves are produced by changing (time variable) electric and/or magnetic fields, it is wrong to picture electric field extending in one direction then a particle or something else pulling it back and forcing it to move in another direction.
This example is more appropriate, take radio waves, these are emitted by rapid acceleration of charges via a strong alternating electric current in an antenna/conductor. The rapid acceleration of electrons i.e. alternating current induces a time dependent magnetic field at some distance, a changing magnetic field has a virtue of inducing an electric field, since the magnitude of magnetic field changes with time so does the value of electric field, the process repeats and wave propagates forward by producing alternating fields one after the other.
This self sufficiency of electromagnetic waves also enables them to propagate in vacuum in absence of any palpable medium. It would further be easier to perceive the propagation if you do not picture electron or other particle moving in any medium as they don't actually, just imagine a variation in time and not in space.
Explanation as required by comments : When I say the fields generate each other one after the other I do not mean that one field collapses and forms the other ! I was referring to the continuum of field generation one after the other, and that each field is a cause for thr other.
A: This is an excellent question! I think the overall confusion arises from descriptions of the EM phenomenon at a multitude of levels of understanding at the same time, each with supporting semantics.
In low-level quantum field theory, there are correlations (in 4D) in the photon field, between electrically charged fermions (like electrons and quarks), and these are always sent out and received by something. I'm hesitant to use the words "virtual photons" but this is where that terminology comes from.
When the distance between the source and receiver is large, due to interference, the only surviving contribution of these (when combined from the right source, see below) looks like the distribution of a classic "photon" with a wavelength/frequency at the speed of light and all that. If you look very close instead, you will see a lot of other strange field correlations between charged fermions that definitely don't look nice and oscillating (and don't propagate at the speed of light either).
So, you could say that the "E/B fields that intermix and create each other", or the springy analogies, are useful ways to interpret math that really arises from events that are fundamentally understood to work at a more detailed level.
To clarify another part of your question - "what oscillates" - the answer is that the quantum amplitude for sending out the field correlations that build up the "photon" oscillates at the source, and this affects the destination. There is nothing "in between" that oscillates. 
It is fair to ask "so why does the source amplitude oscillate in the first place", but this has many answers, and might really require a more specific question. Roughly, think about fermion (electron) field configurations that themselves are of an oscillatory nature. 
Be careful when mixing the levels of view. The various parts of the physics you are looking at have to be described in a compatible way if you are looking for intuition. 
A: My understanding of the wave effect is that the wave is only a representation of a 360° field, consider the charge traveling on a three dimensional graph. The charged wave rotates around the direction of travel. That is to say. The wave rotates in two axis as the direction of overall travel is the third axis. The representation of the a sign wave is the view of the particle as it moves towards or away from you. So a fast climb and slower curvature as it reaches and then passes it's apex, then a fast decline.
Now as to the mechanism of spin that was imparted to the force carrying particle, The best explanation that I am aware of, is that as you apply energy to a particle, you excite it's constituent components. The charge of these elemental components, as they are increased, move faster. The interaction of the separate components starts an interaction that is imparted to the entity as a whole, giving the entire particle a direction of spin. So, the particle moves away from the energizing source and according to the conservation of angular momentum, the imparted initial spin follows that direction of travel around a central axis, directly proportional to the amount of energy applied to the particle by the energizing source.  That is to say, The amount of charge is directly related to the distance of spin around the central axis, as it moves in an overall direction of travel. 
Hope this was clear, And again. this is my understanding of the entire process, if someone can explain it better, I am Grateful for any input.    
A: In general terms as per my understanding. 
an EM wave is generated as follows
imagine a piece of wire of points A to B placed like this  A-----B
when you pass an alternating current (voltage +v 0 -v)
Emission of Electric wave.
E1: lets say for negative voltage's peak value the electron concentrates on one side of wire, lets say it at the end of wire i.e at B
due to charge uneven distribution,
there exist a maximum electric field emerging from A to B, lets indicate this max field to be a + peak of electric wave.
E2: lets say for Positive voltage's peak value the electron concentrates on other side of wire, lets say it at the end of wire i.e at A
due to charge uneven distribution,
there exist a maximum electric field emerging from B to A, lets indicate this max field to be a - peak of electric wave.
E3: when voltage is at 0 there exists no electric field. the electron may not be specifically concentrated any where. so there exits no polarity thus the field is 0.
Emission of Magnetic waves.
(assuming that:  u know how a magnetic field effect is observed when an electron is moving
if not, just google it. its a vast topic to cover here.)
H1: lets say for negative voltage's peak value the electron concentrates on one side of wire, lets say it at the end of wire i.e at B
i.e the electron moment is just about to start to other side now i.e electron velocity is min at this point so there exits min magnetic field and max E1 (as stated above)
H2: lets say for Positive voltage's peak value the electron concentrates on other side of wire, lets say it at the end of wire i.e at A
i.e here also the electron moment is just about to start to other end, here also the electron velocity is min at this point so there exits min magnetic field and max E2 (as stated above )
H3: lets say for voltage in between, the electron in the wire has considerable velocity in it i.e has considerable amount of max current to flow. thus due to this movement of electron, there establishes a magnetic field maximum . i.e here the the field also exists at lower velocity of electron but gets maximum as current is max. here the electric field is 0.
here the generated magnetic field is like your left hand fingers . i.e if you place the thumb in electron moment direction i.e if electron travels from  B to A then the fingers turn in anti clockwise direction and its the direction of field around the wire lets say this direction as + peak value of field and for other moment i.e from A to B you get clock wise direction lets say it -ve peak of H field 
this certain point of max field , in general occurs at phase exactly 90 degrees to that of Electric field i.e H field variation is in phase with current phase and that of electric field is in phase of voltage phase ..
in space this cutting of electric and magnetic fields is also 90 degrees to each other 
i.e like L shape '_' -> electric field '|' -> magnetic filed i.e magnetic field cuts electric field exactly perpendicular to it .. 
when this whole picture is kept in time phase and space quadrature for given A.C input you can get the picture of oscillations of the fields. hope this helps you :) 
A: At absolute zero there is no K.E. of particles in a body but at temperature > 0 K the picture changes as particles start vibrating about their positions . As we know that there are charged particles all around us , they start oscillating and oscillating charges produce Magnetic fields and these Magnetic fields are sine functions so they are also changing , which leads to oscillating Electric fields also in sine wave function . Hence an E.M.W. is formed .
A: Great question! Here is the simple explanation as to why electromagnetic waves oscillate. 
As you probably already know, electromagnetic waves are generated by moving electrons. An electron generates an electric field. A moving electron generates a magnetic field. These combined electrical and magnetic waves reinforce one another, giving rise to an electromagnetic wave. 
What causes the electromagnetic waves to oscillate? The answer is simple. It is due to the fact that matter itself is made up of atoms that are in constant motion within each matter. All matter consists of atoms in motion and these atoms, in turn, consist of positively charged protons surrounded by a cloud of negatively charged electrons. The vibrating motion of the atoms causes the cloud of electrons to oscillate and this oscillation generates electromagnetic radiation. 
Since all matter contains electrons and all these electrons are in motion, all matter generates electromagnetic waves. 
Since all electromagnetic radiation travels at the same velocity, the frequency and wavelength of the generated radiation depends on the frequency of the oscillating electron cloud. Thus, on average, cool objects (say those at room temperature) generate long wavelength (low frequency) radiation, while hot objects (such as the sun) generate short wavelength (high frequency) radiation.
I hope this helps.
A: I don't know what the author of the question thinks, but no one answered his question. He asks why the electromagnetic wave oscillates, the answers explain that these oscillations are induced by the oscillations of other particles, but then, why does the first source which transmitted oscillations oscillate? I would add this: why must all the particles also be waves?
A: First of all When you are thinking light as a classical wave then you can't think it as a photon(particle).light has dual character both wave and particle , and which character will come to us it's depend on our experiment that we will do.
Now when we think light as a wave , then it is special type of wave called electro magnetic wave which is far different from mechanical wave( in which a physical particle is oscillating).
In this case the electric field is varying sinusoidally and because of this variation of electric field, a sinusoidally varying magnetic field created( according to Maxwell law or more preciously Faraday law electromagnetic induction).
So finally this E-M wave is just the space time SINUSOIDALLY varying electric and magnetic fields only, don't confused with classical S.H.O.
Thank you
A: charges  oscillating on a conductor generates EM wave. EM wave can also be generated when charged particles falls from states of high electric potential into states of lower electric potential much like a stone falls. When you detect the wave, what you catch are photons.
