The idea is that the yellow phosphor converts a short wavelength blue light from the LED into a longer wavelength yellowish light, called down-converting.
This is a special physical property of the phosphor material - not simply a function of its color.
Normally this yellow phosphor material is applied directly to the LED die - but they figured they would put it on a plastic shell outside the LEDs. It doesn't actually make any kind of performance improvement except that the light is more evenly distributed.
If you tried to paint the shell with the phosphor - you would most likely have blue light shining through it - that's why they need so much technology to accurately dope the plastic phosphor, so that the light output is an even and consistent color.
The LEDs are typically blue with a phosphor that is energized by the LED light. It's the phosphor layer that starts to glow and produce the white light.
If you "paint" a surface a particular color to alter the light transmitted through it, you're effectively creating a filter. If you want something to appear more yellow, you need to increase the amount of yellow wavelengths transmitted through the filter by reducing the other wavelengths. But if you have a single color LED, such as blue, there are no other wavelengths present (at least not in any significant amount). So if you put a yellow filter over a blue LED, you see nothing...
This is a special physical property of the phosphor material - not simply a function of its color.
Normally this yellow phosphor material is applied directly to the LED die - but they figured they would put it on a plastic shell outside the LEDs. It doesn't actually make any kind of performance improvement except that the light is more evenly distributed.
If you tried to paint the shell with the phosphor - you would most likely have blue light shining through it - that's why they need so much technology to accurately dope the plastic phosphor, so that the light output is an even and consistent color.