In order to understand this claim you have to first understand what causes car sickness in the first place. What is basically happening is the motion you sense with your inner ear is different from the motion you are visualizing. This is easily caused by motions that you sense with your body that you can't see with your eyes or that you can't anticipate. This is most common in cars when people are doing other activity while riding in them such as readying a book or playing on an electronic device. It is generally not an issue for the driver since they are in control and able to anticipate everything.
As to your question the there really isn't any difference in the motion of an automatic car and a manual car as when driven properly there will be no noticeable changes in motion when shifting gears. I would chalk that site up to an advertisement for selling cars with an automatic transmission as it just seems to talk about how an automatic transmission is better then a manual.
If you look around online you will find claims that both an automatic and a manual are better for preventing carsickness.
As a side note there is talk about how driver less cars might make carsickness worse for some people as it will take away the advantages that are gained from driving.