Question

Why can't electric field lines intersect?

Answer 1

Electric lines, at any point, indicate the direction of the electric field. If electric lines intersect, then at a single point, there would be two electric lines, that is, two directions of electric field. But, at one point, there cannot be two directions of electric field. So, electric lines cannot intersect each other.
Similarly with magnetic lines. They indicate the direction of magnetic field. And again, there cannot be two directions of magnetic field at one point.