> The alternative is to mix the light from red, green and blue LEDs, which produces a very peaky spectrum that distorts colors badly

What does the phosphor emit? There's no "white" wavelength. So a more homogeneous spectrum?

Roughly, if the average color of something is white your eyes will make it look white. Since we only have three color receptors you only need red green and blue in the proper amounts. The rest of the visible spectrum is naturally interpolated.

An approximation of white doesn't work for reflections which is the main reason this method isn't ideal. Real world objects can reflect different wavelengths very unevenly so even if these sources appear white directly their reflections take on a variety of strange hues.

The phospor spreads out the peaks so the light spectrum emitted is smoother, closer to black body radiation which is the type of light emitted by the sun. Humans are wired for sunlight so this makes things look better.

Yep. Imagine a spectral histogram. Pure white light looks like this:

     R   G   B
  _=============_

A surface that reflects the wavelength at “x” will look fully saturated and bright:

     R   G   B
  _=============_
           x

But now imagine a “peaky” white. The histogram has peaks at pure R, G, and B, but the other wavelengths are underrepresented or absent.

     R   G   B
    _=_ _=_ _=_

The light will look white to the eye, and surfaces that reflect a wide spectrum will look relatively normal. But look at our hypothetical narrowly-reflective surface from before: It falls in between the peaks, and thus will have a washed-out appearance.

     R   G   B
    _=_ _=_ _=_
           x