Here is an excellent explanation provided by Professor Craig Bohren of Penn State University:

"First I'll give you the wrong explanation, which you can find here and there. It goes something like this. As everyone knows, scattering (by anything!) is always greater at the short wavelength end of the visible spectrum than at the long wavelength end. Lord Rayleigh showed this, didn't he? Thus to obtain the greatest penetration of light through fog, you should use the longest wavelength possible. Red is obviously unsuitable because it is used for stop lights. So you compromise and use yellow instead.

This explanation is flawed for more than one reason. Fog droplets are, on average, smaller than cloud droplets, but they still are huge compared with the wavelengths of visible light. Thus scattering of such light by fog is essentially wavelength independent. Unfortunately, many people learn (without caveats) Rayleigh's scattering law and then assume that it applies to everything. They did not learn that this law is limited to scatters small compared with the wavelength and at wavelengths far from strong absorption.

The second flaw is that in order to get yellow light in the first place you need a filter. Note that yellow fog lights were in use when the only available headlights were incandescent lamps. If you place a filter over a white headlight, you get less transmitted light, and there goes your increased penetration down the drain.

There are two possible explanations for yellow fog lights. One is that the first designers of such lights were misled because they did not understand the limitations of Rayleigh's scattering law and did not know the size distribution of fog droplets. The other explanation is that someone deemed it desirable to make fog lights yellow as a way of signaling to other drivers that visibility is poor and thus caution is in order.
Designers of headlights have known for a long time that there is no magic color that gives great penetration.