How do LCD panels work?
Well, first let's figure out what you mean by "LCD panels." LCD stands for "Liquid Crystal Display." Inserting the word "panel" simply means that the display is in rectangular form (such as a computer screen or even a digital picture frame).
Even though we are most familiar with the states of solid, liquid, or gas, liquid crystals don't really fit into either category. The reason why is because their molecules (the smallest part of an actual substance that maintains its properties) generally stay in the same orientation which is like a solid; however, these molecules are also able to move around which is like a liquid. Luckily, they require a very small amount of power. A negative is that they are quite sensitive to changes in temperature.
Further, the "how" of their workings kind of imitate polarized sunglasses. You know those sunglasses that if you put two lenses in front of each other and turn them, they block all light? That is the essence of of the "how" in regards to how Liquid Crystal Displays work. Because these displays have two polarized layers stacked right on top of each other, they can make use of the twisting property in order to block light or let light through.
A tiny bit of irony is that I am using an LCD on my laptop computer right now in order to see my answer and to send it to you!