Video narration
Why doesn't equal distance from both charges make the field zero? On the equatorial line-the dipole's perpendicular bisector-draw your familiar field arrows. The point is equally far from +q and −q, so both arrows have equal length. Resolve each into sideways and dipole-axis parts. Sideways arrows oppose and cancel; axial arrows both point toward −q. That is opposite the electric dipole moment p, which points from −q to +q. Here is the turn: symmetry cancels only some components. For charges ±q separated by 2a, with r measured from their midpoint to the point, the exact magnitude is k times p divided by (r squared plus a squared) to the three-halves power. Only when r is much greater than a does this become k times p over r cubed. Your arrows balance sideways, but leave a field against p-not zero.