The probability of the emission of a photon to a state where $N$ photons are already present is proportional to $(N+1)$. This fact follows from the symmetry of physics under the time reversal $(t\to -t)$. Why? Because the probability of absorption of a photon, i.e. process going from $N+1$ to $N$ photons in the given state (reduction), is clearly proportional to the initial number of photons which I called $N+1$. The more photons, the more likely the atom is to absorb. By time-reversal (or CPT) symmetry, the emission process has to exist as well and it has to depend on the existing number of photons that are already "out there" because the absorption process clearly depends on it, too.
The probability of emission is therefore proportional to $N+1$. The $N$ part is "stimulated" and the $1$ part is "spontaneous".
The time-reversal calculation above was already known to Einstein before quantum mechanics was found by Heisenberg et al. in 1925. A kosher quantum mechanical derivation of this factor $N+1$ in both probabilities comes from the squared matrix element in the harmonic oscillator
$$|\langle n+1| a^\dagger|n\rangle|^2=n+1$$
or its Hermitian conjugate. It's because the operator responsible for the emission/absorption contains a creation/annihilation operator for the photon.