That's an old jogger's rule that has nothing to do with biking. When you are doing long distance running along the side of a busy street or highway, you are supposed to always run in the direction of oncoming traffic (off to the side of the road, of course). That way you will see the traffic and be able to avoid a drunk or erratic driver by diving into a ditch. If you are running on the same side of the road as traffic is moving, all the traffic is coming from your rear, and too many joggers have been killed or crippled by being hit from behind. When I was a young cross country runner, my high school team always took the oncoming side of the highway in our practice runs.

Bicycling is totally different. You are moving forward much faster than a jogger, and any time you would gain by seeing an erratic driver would be lost by your own forward momentum, and the wreck would be far worse if not fatal. If you are riding on busy streets, ride with traffic and use a mirror. I just rode 15 miles on a country highway today with my mirror - very safe.