First you need some vectors.
If a vector, for instance the force vector, makes an angle of theta to the surface of an object it will include a perpendicular component Fsinthetha, and an tangential or parallel component (parallel to the surface) F cos theta.
The forces are attempting to accelerate the particles in perpendicular and tangential direction in relation towards the earth.
Then, we will return to the issue you had. If a field line creates an angle theta on the surface of the conductor charged the field line will attempt to increase the charge local to it (the charges on the surface in close proximity to the field line) both tangentially and perpendicularly.
The charge is now unable to be faster perpendicularly because other forces are trying at pulling it down, and take the force out.
But the tangential component could increase the charge and could cause them to move across the conductor. However, in electrostatics, the charges are meant to remain in a state of equilibrium.
So, we conclude that when electrostatic conditions are been reached the point where field lines have to be perpendicular to the surfaces everywhere.
If you’re wondering it is only valid for as long as we’re under electrostatic conditions, which you can clearly see by my reasoning.