Because of how the crystals are oriented, and how they block light.
LCs are polarization devices. TN stands for twisted nematic, which describes the physical configuration of the crystals that perform the switching. Nematic means "threadlike" and looks like this:
This results in the liquid crystal no longer being isotropic: now, its optical properties depend on which direction the light is going, and which orientation the light is polarized in.
Twisted nematic refers to setting up a series of these structures so that they point in a slightly different orientation as you go down, the result is a spiral. When in this configuration, they will rotate the polarization of linearly polarized light just a bit with each step down the spiral, through the full twist of 90 degrees (below left):
By putting linear polarizers above and below, you can make it so that the light that passes through the TN structure is aligned so that it will pass through the second polarizer and come out to your face.
The "opaque" state is when the TN structure is disrupted. The crystals are not in their precise arrangement, and do not provide the required rotation of the polarization (above right). The result is that you have two crossed polarizers, and the light is blocked.
You can control the degree to which the TN structure is present, and by doing so change the opacity of the LC cell.
TN's angular issues come from the fact that the TN structure only works if the light is hitting it directly from normal incidence. If you move your head off-axis, you're seeing light that didn't properly go through the TN spiral, and so is only partially affected by the TN structure. This means that any light that goes off to the sides isn't affected by the right level of opacity in the LC cell. With each pixel being made up of individual color subpixels, with each one having a different opacity in order to generate color, each channel is affected differently by moving off-axis, which results in the strange color shifts. In early LCDs, this ended up being a full contrast reversal with only a slight angular offset - more recently, improvements in LC cell design have made it so that this effect is minimized as much as possible.
But it's still something fundamental to TN, something that you can only get away from fully by not using a twisted nematic structure in your LC cell.