The definition of electric field lines greatly eases the visualization of electric fields, a vector field, especially in the presence of many charges. The one-to-one correspondence between the electric field and the electric field lines necessitates that the field lines follow some rules.
For one, the electric field of a positive charge must originate from it. That is because its electric field points away from it. Moreover, since the magnitude of the field asymptotes to zero at infinity, the field lines in the presence of a single positive charge must also extend to infinity.
For a negative charge, the field lines are precisely the opposite. Hence, they come in from infinity and culminate on it.
Since the electric field of a point charge is proportional to its magnitude, so is the number of electric field lines in its vicinity.
By definition, the field line density at any point in space is proportional to the electric field at that point. Also, the electric field vector is tangent to the field line at that point. Now, this implies that electric field lines can never cross each other.
Imagine a point where electric field lines cross. That implies there are two directions of the field at that point. This further implies that a test charge placed at that point would experience a net force that has two directions. Since that is impossible, the hypothesis is ruled out.
