LEDs create light by electroluminescence in a semiconductor material. Electroluminescence is the phenomenon of a material emitting light when electric current or an electric field is passed through it - this happens when electrons are sent through the material and fill electron holes. An electron hole exists where an atom lacks electrons (negatively charged) and therefore has a positive charge. Semiconductor materials like germanium or silicon can be "doped" to create and control the number of electron holes. Doping is the adding of other elements to the semiconductor material to change its properties. By doping a semiconductor you can make two separate types of semiconductors in the same crystal. The boundary between the two types is called a p-n junction. The junction only allows current to pass through it one way, this is why they are used as diodes. LEDs are made using p-n junctions. As electrons pass through one crystal to the other they fill electron holes. They emit photons (light). This is also how the semiconductor laser works.
To understand p-n junctions and semiconductors better you will need to invest a good amount of time in a lecture, it is not a simple phenomenon and far too lengthy to cover here. See a 59-minute introduction lecture to solid-state (semiconductors) here.
Phosphors are used to help filter the light output of the LED. They create a more pure "harsh" color.
Engineers had to figure out how to control the angle the light escapes the semiconductor, this "light cone" is very narrow. They figured out how to make light refract or bounce off all surfaces of the semiconductor crystal to intensify the light output. This is why LED displays traditionally have been best viewed from one angle.