Transverse waves are those waves which vibrate along the axis perpendicular to the axis in the direction of motion. Assume the wave moves along the X- axis then, for transverse waves, the vibrations would be along tge axis perpendicular to X-axis, i.e. Y-axis or Z-axis.
Electromagnetic waves are the combination of electric and magnetic fields vibrating in the direction perpendicular to the direction of propogation of waves. Since the vibration and propogation are perpendicular to each other, therefore, electromagnetic waves are transverse in nature.
Light waves are just electromagnetic radiation of certain frequencies in the visible range. Hence, light waves are elctromagnetic waves. And as electromagnetic waves are transverse in nature as stated above, light waves are transverse in nature as well.