First of all, note that there are two relevant forms of coherence for a laser: temporal (where the phase of the field is steady with time, i.e. the field is single-frequency) and spatial (the phase of the field is consistent across the width of the beam).
Assuming you're thinking of temporal coherence, perhaps try thinking about this classically and starting with a related question:
Why is the light that exits a clear piece of glass coherent (same frequency, phase, and direction) with the light entering?
One way to approach this is to imagine the light field as a time-varying perturbation on the atoms with which it is interacting. Classically, the E-field sinusoidally accelerates the electrons, getting absorbed into their motion. When those accelerating charges re-emit the radiation, it will be with the same phase, frequency, and polarization as the exciting field because that's just how the charges happen to be moving--they were driven that way by the field. Thus the light emerges the same color and phase it had going in, and no one is surprised.
Stimulated emission is a similar situation. While (obviously) more complex in certain ways, again one can add a sinusoidally-varying field to the electron Hamiltonian. In a population-inverted gain medium, this will have the effect of causing the electrons to evolve to a lower energy state, emitting light. What color and phase will the light have? Well, the same as the excitation light since that's just how the electrons happen to be moving. There is simply no other phase and frequency for them to have, because there is no intermediary between them and the driving field; they are directly driven.
All of this is to say that any process where the emission is directly driven by an external field will preserve the phase information. These are the coherent processes, including, for example, nonlinear second-harmonic or difference-frequency generation. These are in contrast with incoherent processes, such as photoluminescence, which are basically forms of spontaneous emission. Incidentally as well, preserving the frequency information of the exciting field is a matter of conservation of energy, since photon energy equals frequency times Planck's constant.
So this just describes stimulated emission. In a laser, you have feedback, the dynamics of which add another layer of complexity and result in the formation of a single dominant state existing in the cavity. A topic for another day, I suppose.