it's not unusual to find a single turn potentiomenter (aka pot) that provides a change in resistance over the range of the turning component. The benefit of a single turn pot is that you can get a large change in resistance with minimal movement. Volume controls are a good example.
A multi-turn potentiometer is just the opposite. It provides a similar range of resistance, but requires that you perform multiple rotations of the control to "travel" from one end of the range to the other.
As an example, a volume control may provide 10K resistance from one point to the other. You can select a 5K resistance (5000 ohms) by approximately turning to the halfway point. Due to the large range of resistance over such a small movement, it is difficult to pinpoint the 5000 ohm figure. For something as objective as volume control, such a coarse level of resistance is acceptable.
On the other hand, with a ten turn pot, as in your example, of the same 10K value, you can easily guess at five turns to get close to the 5000 ohm level, but more important, you can move a quarter turn and have substantially less change than with the single turn pot and be in a better position to get the figure you require.
This allows one to provide for fine adjustment of various devices, such as the servo mentioned in the article you've linked.
If you had a circuit to control a servo, using a single turn pot, you may get 180° movement over the full turn of the pot, but with the ten turn pot, each turn would possibly provide only 10° of movement.
This is a generalization that I'm hopeful will assist your understanding.
You ask how to modify a servo with a potentiometer. A servo is a small motor with a potentiomenter within the housing. The potentiomenter sends a feedback signal to the controller to inform the controller of the location of the servo. What modification do you require of this internal pot?
Some servos are engineered to turn more than 90° or 180° and some can be modified to do so. This would be considered outside the scope of this answer and also can be found by using search terms such as "modify servo rotation" or similar. You may also discover such a reference in the world of Instructables such as this provided link.