What is the purpose of logo lights on airliners?
There are multiple reasons.


First of all: Of course promotion for the airline.
Another reason is the visibility. Airliners have several lights and strobes (Nav lights, rotating beacon, ...) but this can be overseen. The logo light is used to see if there are any other aircraft crossing the taxiway or apron. It's just better to have a big, bright illuminated area, instead of a few small lights.
And ATC has also something to do with it. Many phrases regarding taxiing include airline names. For example:


  AFA739, give way to an Air Berlin A320 moving from left to right on taxiway Bravo


ATC and the pilot can visually identify the companies name thanks to the logo lights.