Question

Why the electric lines of forces terminate on the conductor normally?

Answer 1

Within the conductor.
Since it is a conductor :).
If there would be a net electric force acting on the electrons within it thus moving them until the force is neutralized. Therefore (after a short period of time following the introduction of charges on the conductor) any electric force within a conductor is neutralized.

Answer 2

if they were not normal to the surface, there would be a component along the surface whish would cause the charges to mose. If the system is in equilibrium (electroSTATIC), then there cam be no component along the surface