What they mean by "chew your juice" is to just let each sip linger in your mouth for the amount of time it would take to chew it. It is liquid, and yes, you could simply swallow it. But if you leave it in your mouth for a few moments, it gives your saliva a chance to start digesting it.

Another thing to consider is the foam. Apples, in particular, seem to create a lot of foam for me when juicing. Try to remove as much of the foam as you can before drinking (chewing) your juice. This will help with the bloating and gas.