On March 4, 1829, Andrew Jackson was inaugurated on the East Front of the Capitol (the side that now faces the Supreme Court building). With rare exceptions, that was the site of regularly-scheduled presidential inaugurations from then on, up to and including that of Jimmy Carter on January 20, 1977.
Then, when Ronald Reagan was inaugurated four years later, a custom began, which has been followed by all of his successors, of holding the event on the West Front of the Capitol.
I don't know that there was any one reason for the change. I've variously heard that 1) more spectators can be accommodated on that side; 2) it's a nicer view, looking over the Mall, rather than a parking lot; and 3) Reagan preferred the symbolism of facing west toward most of the country, rather than away from it.
Whatever the reason, everyone seems happy with the change. This year's inauguration will also be held on the West Front.