Why do electromagnetic waves oscillate?

If you have a charge, it has an electric field around it. If the charge moves, it appears that a magnetic field is also present at a given point.

Aside:
Assume that you have an infinite length of charged wire. It exerts some force 'F' on a charge 'q' at a given point. This is electrostatic force, which can be derived using Coulumb's Law.

Now, you start moving parallel to the wire. The charge appears to go backwards. Per Special Theory of Relativity, the length contracts (same charge in condensed length) and time dilates (momentum of charge changes), and the apparent electrostatic force on the charge changes. But, the force on the charge should remain the same, as your change of reference frame should not have any impact on force. The additional force is 'magnetic force' caused due to 'current'.

So, essentially, electro-static force and magnetic force is just a single force, called 'electro-magnetic force' - manifest as different forces depending upon the reference frame.

So, a changing electric field causes changing magnetic field, which in turn causes changing electric field.

These two fields happen to be perpendicular to each other (you can see that in the above example. Electro static force when charge stationary repels it with some force 'F'. You can derive the magnetic force magnitude and direction from this thought experiment).

A charge oscillates. It creates a changing electric field, which induces changing magnetic field which in turn induces changing electric field, and so on.

This whole phenomenon spreads out in space and that is called as 'Electromagnetic radiations'.

So, you see, the radiations themselves do not oscillate. What oscillate are electric and magnetic fields.