The color a LED emits is given by the choice of materials (specifically, the band-gap, with E=hν, ν being light frequency https://en.wikipedia.org/wiki/Band_gap#Light_emitting_diodes... )

"Tunable" LEDs are just multiple LEDs, the goal being to change the emission peak. Peak wavelength can be mapped onto a black body temperature trough Wien's law (λpeak =2.8e-3/T) https://en.wikipedia.org/wiki/Wien%27s_displacement_law

I don't really know how the mapping is done, though, as there are different ways to map it since the LEDs won't have the same emission curve as a black body.