As said THANATOS the time dilation it is the experimental fact (#3). When particles are moving faster then live longer. The real problem is why they live longer. I think that problem is because we are talking too much about the mathematics, not about the physical meaning of this effect.
A space is more curved when the energy density has higher gradient i.e. has higher the energy density drop. It means that the more powerful vortices (cyclones) give the more curved spaces. In such space we can move between two points, for example equidistant of the centre of a vortex, only along an arc. If the curvature is higher then the path is longer and a light needs longer time to reach the second point. It is dilation of time. As said Einstein, in spaces of higher gravitational field intensity dilation of time is also higher. When we accelerate some object then the angular velocity increases (the object rotates more quick) – it means that there is created stronger vortex and it means that in such object time is going slower i.e. the separation of particles in the decay, for example, of the muon last longer because the particles are moving along the more curved trajectories.
What about the twin paradox. The Einstein special theory of relativity and the Bohr quantum mechanics are correct but incomplete. This incompleteness was the source of permanent quarrels between these genius scientists. Einstein to the end of his life did not understand his statement that light has speed “c” in relation to all objects in space (true) simultaneously (false) (see #26, thread ‘Before the Big Bang’). The assumption that the Universe is built of one component only, i.e. the neutrinos leads to the wrong interpretation of the relativity theory. The existence of the quantum mechanics proves that there must be at least one component which moves with the velocity much greater than the light velocity in vacuum because from the quantum mechanics results that in our World there do not exist systems separated spatially. How one should interpret the relativity of time or relativity of the clock rate? Imagine that in the whole Universe there exists only one spherical object and connected with it the set of the eterion-higgson streams. At certain moment the object explodes and decays on two identical parts. We know that the velocity of each part in relation to the other is the same. In which part the time goes slower? Of course the time goes in both parts the same because we have here a perfect symmetry. Now we can explain the twin paradox. Twin brother who moves in rocket in relation to the eterion-higgson streams connected with our Earth will live longer because this streams are stronger than created by rocket.