why is it that only the perpendicular component of a wave can pass through a vertical polarising filter? To me it seems unintuitive that a filter can change the direction of a wave. I also do not understand malus' law, could somebody please show me an intuitive derivation or proof of malus' law.
Is unpolarised light, different waves with different oscillation directions, all coming from the same source?
 A: The component of the field that is parallel to the wire looses its energy in the wire by Ohmic dissipation so that only the perpendicular component remains.
Given this, only the perperdicular component $E\cos\theta$ of $\vec E$ will go through.  Since the intensity is proportional to the square of the magnitude of $\vec E$, you get $I(\theta)=I_0\cos^2(\theta)$.
A: 
To me it seems unintuitive that a (polarization) filter can change the direction of a wave.

What is influenced by a polarization filter?
What you are calling a wave is light emission or electromagnetic radiation. And it is only a small range of the EM radiation from all the existing wavelengths. A well designed polarization filter let through a well defined range of light, say from blue to red. EM radiation of infrared as well as of X-ray ranges or not going through or going through nearly uninfluenced.
Light from a thermal source consists of photons with their electric and magnetic field components. The directions of this field components are randomly distributed in space, means this light is not polarized.
From the fact that filters are not only of a well determined slit width but of a finite dimension of the slits length (for example a filter for a lens could be of a diameter of approx. 20 mm) one could conclude that the photon in practice has an extension which is influenced only inside a well erudable distance:


*

*Be the wavelength of the photon smaller - in comparison to a photons wavelength which is influenced by a slit of well defined width - it can goes through not influenced.

*Be the slits length of a finite extension this doesn't influence the photon despite it is stated that fields from a source are extended to infinity.


What is not influenced by a polarization filter?
All photons which are detected behind the slit(s) are remaining of the same wavelength. This can be proofed with monochromatic light which indeed stays of the same wavelength behind the slits. This holds too for polarized light which goes through a second polarizer.
The last needed experiment -this time with three polarizers with orientation 0°, 45° and 90° - gives the result, that light is going through although in the case of the removed second polarizer no light is going through the crossed polarizers.
Conclusion: The photons field components are rotated under the influence of slits despite the photon wavelength/frequency/energy stays unchanged.

Is unpolarised light, different waves with different oscillation directions, all coming from the same source?

Yes, if you understand under a source an electric bulb or a laser or a LED. No, if you are going in detail and understand under a source the electrons from which the photons are emitted.
