Intuitive explanation to why force perpendicular to velocity results in circular path
This is not always true, sometimes it may result in a circular path given the boundary conditions and the forces involved.
In short, my question is exactly how and why a stone attached to a string follows a circular path when its velocity is perpendicular to force?
It is a fact that turning a tied stone over your head ( to keep the circle in the same gravitational field for simplicity) will result to a circular path. The circular comes because in your experiment the string gives a maximum radius for anything tied to it ( even a dog) and a circle is defined by r=constant, maximum distance.
So this part of the question reduces to why is the string at maximum distance, which is the second part
Also, why does not the stone falls into the center because there is a force pulling it towards the center?
Lets try it in turn.
Newton's first law says that a body in an inertial frame stays at rest or moves with constant velocity if no force is acting on it.
Take an infinitesimally small interval of the path of the stone, in that interval the stone has velocity (three vector) v and can be considered to be in an inertial frame, so it should fly away on the tangent ( breaking a window?), why does it not? Because a force is applied in a Δ(t) that changes its direction, pulling it in, so it is no longer in an inertial frame. This force is applied by you through the tension in the string ( and the electromagnetic forces that hold the string together). If the string is cut the stone will fly off on the tangent, the stone following its instantaneous inertial frame, since no force would be applied to it.
The force that is pulling it towards the center while tied is working against the instantaneous impulse (dp/dt) for the stone to leave on a tangent ( breaking the string). These are called the centripetal and centrifugal forces equal to each other , the centrifugal an apparent force coming out of the mathematics.
If the force you apply through the string is not enough to keep the stone circling over your head , it will not go into a circle, but a random fall, due to gravity.
One does not need a string to generate circular tracks. These bubble chamber tracks coming out of a gamma annihilation into electrons and positrons are in a magnetic field, which interacts with the motion of charged particles .
The two spiralling tracks in this bubble-chamber diagram were made by an electron and a positron. These particles were created by a high-energy gamma ray in a collision with the electron of a hydrogen atom in the bubble chamber. The long slightly curved downward track was made by the recoiling electron.
The force conducted by the string, is substituted by the $Bqv$ of the magnetic interactions giving the centripetal force, while the centrifugal $mv^2/r$ balances into a circle of radius r. The radius is diminishing in these pictures because the ionization which makes the tracks visible reduces the velocity of the track. All high energy physics data depend on this effect on charged tracks in a magnetic field.