Why does this part of an OLED filter light only when orientated correctly?
How does the tinted film only filter in one direction?


That is an OLED screen, not LCD, so the structure is a little different.  Instead of having just the top polarizer, an OLED screen has the top polarizer attached to the top of a quarter waveplate:



When oriented the correct way, the polarizer converts incoming ambient light to be linearly polarized.  The waveplate then converts it to circular.  After reflecting through the display, the light passes through the quarter waveplate a second time, accumulating 1 half wave of retardance.  This turns it back into linear polarized light, but with the perpendicular polarization, causing it to be absorbed by the polarizer.

In the incorrect orientation, the light hits the linear polarizer before the waveplate and is not rotated one half wave.  Without that rotation, nothing is absorbed, which is what you observe. 


  What is the name of this effect?


It is sometimes informally referred to as the poor man's isolator because it can be used to mimic part of the effect of a Faraday isolator (allowing light to pass in one direction but not the other) but without the cost, weight and magnetic field. Aside from that, I don't think it has a name, or is even considered to be one specific effect. It is simply due the fact that order matters in this particular sequence of filters.


  Is there a simple explanation of how linearly polarised light can be converted to circularly polarised?


Not really.  It also isn't very important here.  The key point is that passing through a half waveplate will rotate polarization by 90 degrees (vertical becomes horizontal) while the polarizer absorbs one state and passes the other (vertical passes, horizontal is absorbed for instance).  Passing through a quarter waveplate twice adds up to one half wave, so ambient light hitting the screen that reflects off the display panel will pass through the quarter wave plate twice, be rotated 90 degrees, and then (if the plate is in the right orientation) hit the polarizer last and be absorbed.  It doesn't matter what the polarization state is in the middle. 


  But then, how do they ensure the material produces a 1/4 wavelength delay? Does it have something to do with thickness, but then white light is made up of all different wavelengths?


If very high rejection of light is required, a waveplate can be achromatized by taking two waveplates made of different materials and combining them.  By selecting materials with very different dispersions and setting the ratio of retardance between the two materials correctly, the change in refractive index in each material can cause retardance to be nearly constant over a range of wavelengths. This is usually not so important for what is basically just an anti-reflection coating though.