By definition: If a positive unit test charge is placed on a particular electric field line, the line points in the direction of acceleration of test charge. Force experienced by unit test charge is proportional to the strength of the field.
Let us move a step ahead of the definition stated above. We need to remember that these lines are tools to visualize and understand vector fields and as such do not exist physically like lines on a Globe showing the longitudes, latitudes etc.
Figure depicts the electric field created by a positive charge, negative charge, and an uncharged object.
Each point on the electric field line defines a unique vector which tells us the direction and magnitude of the force experienced by the positive unit test charge. The tangent to the path at each point must be parallel to the line of electric field vector at the point of observation to define the direction. The density of lines depicts the magnitude of the electric field vector. The number of field lines in the perpendicular unit area at the point of observation, being proportional to the strength of the electric field.
After having stated how we define an electric field line, suppose now that any two electric field lines intersect at any point in the field.
Lets find out the direction and magnitude of the vector at the point of intersection to ascertain the force felt by the positive unit test charge.
We can not. As the field line point in two different directions here. Nor can we find out the number of lines of field in perpendicular unit area at the point of observation. It implies that the supposition is not consistent with the definition of the electric field.
Same argument can be extended to the case of multiple charges having different signs and magnitudes. In this case we need to calculate resultant field vector due to all the charges present. The resultant is going to be a unique vector at the point of observation.
I am sure; whenever you wondered why don't longitudinal lines on a globe, howsoever close these may be, cross you had your answer ready.
Do still think that Electric lines of force could intersect?