I think you have to use, at a minimum, the concepts of 1) force, 2) force changes speed 3) one force can be imagined as two different forces into two perpendicular directions.
Staring with that, you can watch the pole from the top. I'll leave the details to you, an concentrate in the basics steps, using a figure like the one below.
a) The cord always makes a force towards the center
b) Look at the position of the ball just before it reaches the "horizontal" position. Then, one component of the force changes the speed in the vertical direction, the other in
the horizontal direction. Then, the speed "upwards" increases and the speed "outwards" decreases.
c) When the increase in speed upwards matches exactly that of the increase inwards (or decrease outwards), then the movement looks circular because the net effect is a change in total speed direction, but not total magnitude.
d) If you push the string inwards (the details doesn't matter, you can either stay at the center and actually push it in, or let it wrap around the pole), then the force increases and the two components no longer match each other to keep the total speed constant. The balance is lost and the ball will not move in a circular path, but will move both, inwards and upwards in a way faster than when the motion was circular. So the ball motion "upwards" becomes faster than before, and also there is a net motion inwards and the ball gets closer to the center.
Well, after I wrote this, I realize that the picture given is still a little inaccurate and the explanation more complex than I was hoping. But I hope it still could help.
Warning: Copyright infringement. Part of the the picture was stolen from Wikipedia without permission. Do not attempt this at home, Federal charges could apply.