Suppose you want o find field at a point P which is at a distance d from plane. Drop a perpendicular PO from P on the plane. At O make an axis (say x-axis) perpendicular to direction of current. Now take a thin strip of current carrying conductor lying between x and x+dx. It carries a current of idx. For every such strip there will be another strip llying between -x and -(x+dx) which will also be carrying equall current. Now if you want to find resultant of these to field, you will find that component perpendicular to plane will be cancelled while that along x will add up. So Net field will be along x-axis.

 

This is true for a strip at any x. So taking all strips from 0 to . We find that net field at P is parallel to x-axis