An electric field can be illustrated by field lines. The field lines indicate in which direction the electric force would act on a test charge if the test charge is placed at some location exposed to an electric field. That is, if we "place" a charge at rest "on a field line," we know exactly in which direction it would move.
Let us assume that two field lines DO cross each other in a point in space. Now place an electric charge at rest at this point of intersection. In which direction will the charge move now? Along one field line or along the other? The direction of motion of the charge would be ambiguous in this case! The charge cannot move along both field lines at the same time. The charge would have to 'decide' at the intersection point along which field line it wants to move. Exactly therefore it is not possible that the field lines intersect in any point.