In short, to keep oxygen out.
But there are two specific reasons why wine bottles are stored on their side: Oxygen in wine & the wine cork.
Oxygen is the worst enemy wine has.
During winemaking it is made sure that their is no oxygen present in the wine tank after fermentation is complete, during bottling and in the bottle.
A device has been created to assist in the bottling line to keep the oxygen out. A cork is then placed on the neck of the bottle.
The bottle is laid on its side to keep oxygen out.
Wine Corks are used to seal off the neck of the wine bottle since it can expand to its surroundings.
When wine bottles are laid down on their side, they keep the cork moist.
This reduces the chances of the cork drying out and creating pockets that would allow oxygen in.