For the moment, ignore the 240V situation and just think about one hot leg and the grounded neutral. If you were to put a scope on the hot wire (don't really do this!), you would see a sine wave traversing both above and below the 0V center line. Now add a resistor to your mental model. On the positive half wave, the current flows in one direction through the resistor and on the other half wave, the other direction.
Now, for the 240V situation. As you stated, the sine waves are 180 degrees apart. Mentally color one red and the other yellow. At some point, the red is positive with respect to the yellow and our resistor (yup, we had to wire it up again) sees current flow in one direction. A short time later, the yellow wire is positive with respect to the yellow wire and the current is flowing in the other direction.
Pretty simple when you color the wires!