The active ingredient of Dettol is a chemical called PCMX or to give it's full name, 4-chloro,3,5-xylenol. This phenolic compound has quite a few other names as well.
PCMX is only very slightly soluble in water, but it is fairly soluble in many organic solvents such as Isopropyl alcohol (Propan-2-ol). The Isopropyl alcohol itself is extremely soluble in water.
During the manufacture of Dettol, the white crystals of PCMX are dissolved in isopropyl alcohol, which is essentially the main solvent. Other water insoluble compounds are also added to the mix such as pine oil etc which act as carriers for the PCMX and have antimicrobial activity in their own right. Dettol can be therefore considered to be a solution of water insoluble compounds in a water soluble solvent.
When Dettol is added to water, it immediately forms something called a nanoemulsion. Now emulsions are usually either of the water in oil type or the oil in water type and in this case, it is the latter. The Dettol nanoemulsion is composed of extremely small oil particles (containing the PCMX) suspended in the water. To give some idea of the size of these droplets, they are smaller than the majority of bacteria.
The particles in the nanoemulsion are so small that they possess the ability to scatter light. Now, our ability to see colours depends on the amount of absorption or reflection of light in or on a given surface. When an an object appears white, it's because no absorption of light has occurred. Because the Dettol nanoemulsion contains particles that scatter or reflect light, it appears to be white.
Incidentally, unlike many ordinary emulsions, nanoemulsions generally never separate. Another example of a nanoemulsion is milk and once again, it's the reflection of light from the nanoemulsion that gives it its white colour.