Why is it counter intuitive that for a particle moving in a circle is always being acted upon a force which is directed towards the centre of the circle and not towards the direction of the motion of particle?
Look at it the other way round: if there's no force directed to some center point, a body will not move in a circle.
If you don't apply a force to a body, it moves in a straight line. Imagine a stone on a flat, slippery surface. It won't move in a circle.
So, if you observe something moving in a circle, there must be some force dragging it to the center. Imagine letting a stone at spinn around you, helfd by a string. It will only follow its circular movement as long as you pull hard on the string (apply a force directed to the center).
If you apply a force in the direction of the movement, the body will accelerate. On earth, often friction on the ground interferes, so again imagine a rock on a slippery surface.