Temperature is closely connected to the kinetic energy of particles that make up stuff, so the first thing that we must say is that heat is actaully a sum of kinetic and potential energy of these particles( i stand corrected, heat is the part of the energy that is transfered because of the temperature difference). This energy is stored in a chaotic motion of these particles and in a potential form due to intermolecular and interatomic forces that make up chemical bonds which hold stuff together. This has been proven experimentaly so we do know that heat is just this motion. Transfer of heat will happen if one body is at the higher temperature than some other body to which the energy is being transfered. In this context temperature tells as where the heat flow will go....experimentaly this has been proven. So we can imagine particles that move at higher rate exhibit higher temperature. If we now imagine particles that are totally at rest what is their temperature?
Well, these particles have zero heat and zero ability to transfer heat, so this can be called the lowest temperature possible. This lowest temperature possible is also called absolute zero as you said. So we did not know thateverything stops at some temperature we knew that motion is related to temperature and that no motion means no heat transfer. It is only logical to call this no motion state absolute zero. If you than define some temperature scale you can put a number on this zero. Extrapolation is a procedure by which, by means of examination of part of some curve, we predict the behaviour of that curve in the range of values of variables that are not yet accesible. In this manner you can plot the curve for the pressure of the ideal gas against temperature and see where it gives zero pressure and conclude that it means zero motion, and by the logic mentioned above, zero heat transfer id est, by definition, zero temperature.