How can the centripetal force lead to objects flying apart? I don't understand how the centripetal force, which always points to the center of our circular motion can cause this scenario:
We have a big stone which spins very fast, so fast that a part breaks down, because of the centrifugal force (this is at least how my text books describes it). 
My problem: the centrifugal force does not really exist (we only use it in accelerated frames of reference, so the newton laws still work there), so if we're in a laboratory frame of reference, which force would "pull" that piece of the stone to the outside of the circle, when we only have the centripetal force (as mentioned pointing to the center of the circular motion...)? 
(Please don't try to explain it in an accelerated frame of reference, because there I understand it, but I don't understand it in a laboratory frame of reference)
 A: You seem to ask two different questions:


*

*Why does the stone break apart? For the smaller parts of the stone to travel along circular paths, a centripetal force must be exerted on each of them. These forces are exerted through the forces holding the parts together. When these forces become too large for the particular material, the stone breaks into smaller pieces.

*What makes the parts fly outwards after the stone has broken apart? Since there is no longer any centripetal force making the parts travel along circular paths, they simply continue traveling straight in the direction they were traveling before the stone broke apart. This means that they will fly outwards, although not straight radially.

A: In the lab frame of reference, you need to reverse the question - don't ask yourself what pulls the particles apart but what keeps them together.
By Newton's laws, everything on which no force acts keeps travelling in a straight line. So what requires explanation is not that a collection of moving particles - such as a rotating flywheel - flies apart but what keeps them together. The force that keeps them together is a centripetal force, in this case exerted by the bonds that keep the material together. When you reach a velocity where this force is not enough anymore to keep the particles on a circular trajectory/bound orbit, they fly apart.
A: The centrifugal effect is not a force, true, but it does indeed exist.
The effect happens, not because any force pulls the piece outwards in the circle, but rather because of the already existing centripetal force that pulls all the rest of the material inwards in the circle. The piece wants to continue its straight line motion, as everything does when not experiencing forces, which brings it away from the circle. 
If you sit in a car and feel squeezed to the side as the car turns, it is not you being squeezed into the side, but rather the car squeezing into you; turning and trying to take you with it. 
This tendency of wanting to continue in a straight line while being pulled around in a circle, is what we can call the centrifugal effect (I deliberately don't call if "force" to avoid this confusion). 
A: It might help to note that using polar coordinates, at the instant a part breaks away, the part only has a tangential velocity and zero radial velocity. The break away part is not moving "outwards", but instead "forwards" (absent gravity and drag, there would be zero net force on the part). After some amount of time, that forwards velocity does result in the part moving away from the center of the spinning stone.
A better example of centrifugal force in a non-rotating frame is the reaction force in response to a centripetal force, a Newton third law pair of forces, but each part of the pair exerts a force on the other object. In the case of a string exerting a centripetal force on a stone, the stone exerts a centrifugal reaction force onto the string. Wiki has an article about this:
https://en.wikipedia.org/wiki/Reactive_centrifugal_force
There is an example where the only forces are centripetal, a "two body" system where two objects are orbiting each other about a common center. Each object experiences a centripetal force towards the common center (which is also towards the "other" object), but since the force is gravity, there are no reactive forces. In the case of gravity, the Newton third law pair is the gravitational force each object experiences.
A: Try to imagine, instead of a big stone, a big plate. On top, you fill the plate with sand. Now you start spinning the plate.
What's going to happen to the sand? The sand is going to leave the plate very fast, and spill in all directions. This is the basic, natural state of things, and it's from here that you should start questioning.
How do we keep the sand from leaving the plate? The answer is the centripetal force. If you tie each sand grain to the center of the plate with strings, when rotating each string will pull on its grain of sand, and prevent it from leaving the plate.
Using a big rock is the same thing; instead of using strings you're just relying on the intrinsic cohesivity of the rock with itself. Here each "grain of sand", or piece of the big rock, is attached to the pieces adjacent to it. They in turn transfer all forces to the ones adjacent to themselves, and this is how the centripetal force is propagated from the outside to the center of the big rock.
Once any piece of the rock fails to sustain this force, the bonds break. Once again you have sand, and as we said above, sand is going to shoot in all directions.
A: You don't have to explain this by centrifugal force, or any fictitious force at all. All what centrifugal force is about is inertia.
As your stone is spinning, it has some velocity. But since initially there's a centripetal force, this velocity constantly changes towards the center of rotation. When part of the stone breaks off, it's no longer held by centripetal force, so it just flies due to inertia with constant velocity.
It's just when you go to the frame of reference associated with the stone, only there you get centrifugal force — as a device to make Newton's laws look unchanged.
