Step 1 (What a field line stands for): A field line is drawn so that a positive test charge placed on it would move along the line; its direction at each point is the direction of the force (and hence of \(\vec{E}\)) on that charge.
Step 2 (Test at a crossing): If two lines met at one point, a test charge placed there would be told to move in two different directions at the same instant.
Step 3 (Physical impossibility): A single charge at a single point can feel only one resultant force and therefore move in only one direction. Two directions at once is physically meaningless, so the crossing cannot exist.
\[\boxed{\text{Unique field direction at each point} \Rightarrow \text{field lines never cross.}}\]