Try envisioning it as though the two points of the circular needle were NOT attached via the cable. You would start off with your cast on stitches and working yarn on the left-hand needle of the circ and start knitting onto the right-hand needle just as you would if you were working on straight needles. Then, once you've worked all the stitches onto the right needle, you will turn your work around, exactly as you would with straight needles, and begin working onto the empty needle. Does that make sense??
The reason that a circular needle would be called for in a pattern could partly be preference of the person who wrote the pattern, or in some cases, for example when making an afghan that has a lot of stitches, it can be very difficult to fit a large # of stitches onto regular straight needles and the work can get very heavy and is difficult to support with the arm and wrist. When working with circular needles, the weight of the item is distrubuted much better and it puts a lot less strain on your body.
Hope this helps!