Electric field lines represent the trajectory/path traced of a charged particle placed in the field due to the force experienced by the particle due to the field.
Now, if the field lines would have been intersecting, the motion of any charged particle at the point of intersection would have two directions which is impossible and absolutely absurd.
In other words, the field line represents the direction of the field at any point. At the point of intersection, it would imply two directions due to the presence of two field lines at the point of intersection. This never happens and is completely absurd.