Suppose you are short the index option, and long the single stock options (all vanillas). You size it in such a way that at inception you have flat vega, you hedge out all your deltas.
Now assume the market moves down. All your options move away from ATM and they all have less vega (both your long single stock options, as well as ur short index option).
OTM options are long vega convexity: when implied moves up, vega moves up.
So while all your options lost vega due to moving away from the money. They will gain incremental vega due to a higher vol (assuming vol moves along the smile).
Index smile is steeper than single stock smile. So the index option - which you are short- gains more vegas relative to the single due to vega convexity. Combining it all, you are short vegas now. (both index and single lost vega due to spot move, but index lost less due to steeper smile).