If a Pokémon with Illuminate is the leading party Pokémon, wild Pokémon are twice as likely to appear.
sumwun is perfectly correct in his concise summary of shiny-non-shiny ratio being unaffected. That is to say, the chance is still 1/8192 for every Pokemon you encounter.
If you're interested in statistics, then you can see how that means you have a greater chance of encountering a Shiny Pokemon due to the sheer increase in the number of trials, as opposed to the probability of the event itself. If you are not much into that, I can interest you in a George Bernard Shaw quote where he says
When I was young, I observed that nine out of ten things I did were failures. So I did ten times more work.
Between you and me, though, if you are interested in getting a shiny Pokemon, Illuminate does help out; just not in a direct manner. The chance is still the same.