The average value is zero because both fields oscillate as sine and cosine waves which, over one period, has average as zero.
But you also have to understand that the negative portion of these waves delivers power just as the positive portion of these waves. This means we really are not interested in the average value of the fields but the root mean square of the fields.
The root mean square finds the average value of your waves once the negative values are normalized to be positive. For sin and cosine waves this results in roughly 0.707
So you can see, although the waves may oscillate around the zero point, they provide real power instantaneously when they are not zero, regardless of positive or negative values.