Why An electromagnetic wave is transverse: the electric field and the magnetic field are at right angles to each other, and the direction of propagation is perpendicular to both of them. This is described by the Poynting vector, the cross product of E and H, which points along the direction energy flows. So if the E field is vertical and the H field is horizontal, the wave moves straight out at 90 degrees to both.
Watch out The idea that the wave travels parallel to the fields describes a longitudinal wave like sound, not an EM wave; radio waves have no field component along the direction of travel.
Think of your right hand: E, H and direction of travel are three mutually perpendicular axes, like the corner of a box.