I have heard that it is because they absorb electronically in the ultraviolet range, if this is true, why is it true. What is it about the nature of organic molecules that make them absorb in the ultraviolet range.
Most substances are white in powder form if they have no strong absorption of visible light. They whiteness is a consequence of light scattering by small particles and the fact that there is no absorption within the visible range.
The lack of visible absorption is another issue. Light is absorbed by molecules when the energy difference between a filled orbital and an unfilled one in the molecule matches the energy of a photon of light. For light photons the energy is the equivalent of wavelength and therefore colour.
In most small, simple organic molecules, the lowest unfilled orbitals are a long way from the highest filled ones. So they need a lot of energy to cause them to jump to an unfilled orbital. So they absorb only high energy photons (UV light is higher energy than visible light so this implies the absorption bands will be in the UV and we will not notice them as our eyes can't see UV).
The characteristics that create colour in molecules are often long runs of unsaturated bonds where the electrons can spread along the chain of unsaturation. In these delocalised systems the gap between occupied and unoccupied orbitals is smaller than in simpler molecules and often falls into the visible spectrum. These molecules are perceived as coloured.
The molecule below is a simple example of a dye (most are bigger and more complex than this):
The electrons in this molecule can delocalise over all three rings via the NN bridge and this creates orbitals with the right energy gap to absorb blue light.
In simpler molecules with little delocalisation the energy gap is far into the UV. Even in polycyclic aromatic systems the energy gap is often still in the UV.
Of course, this is a simplification, but the essence of what sort of molecules you need to cause visible colour is a relatively large unsaturated system and by definition this doesn't happen in very simple organic molecules. Wikipedia lists many more examples of dyes so you can get a feel for what sorts of structures make strongly coloured compounds.