The field lines indicate in which direction the electric force would act on a test charge, if this charge is placed in the electric field "generating" the field lines. For an electric charge at rest, it would mean that it would start moving in the direction of the force.
Suppose that two field lines cross at a point in space. An electric charge is now placed at this point in space. The direction of motion of the charge would then be nonunique. It would have to move along the field line. But at the crossing point there is a "branching" (just this non-uniqueness of the direction of force). The charge would have to 'decide' at the crossing point, along which field line it wants to move. Exactly therefore it is not possible that the field lines intersect in any point.