How did Einstein know that it was necessary to include "stimulated emission"? How did Einstein know that it was necessary to include "stimulated emission" which was in the same direction as the incident photon and in phase with it?  
 A: Strangely enough even if he did not like quantum mechanics Einstein did model stimulated emission within a statistical framework:

Einstein devised an improved fundamental statistical theory of heat, embracing the quantum of energy.
Albert Einstein first broached the possibility of stimulated emission in a 1917 paper, having turned his attention the year before from general relativity to the interplay of matter and radiation, and how the two could achieve thermal equilibrium. Einstein devised an improved fundamental statistical theory of heat, embracing the quantum of energy.
First, Einstein proposed that an excited atom in isolation can return to a lower energy state by emitting photons, a process he dubbed spontaneous emission. Spontaneous emission sets the scale for all radiative interactions, such as absorption and stimulated emission. Atoms will only absorb photons of the correct wavelength: the photon disappears and the atom goes to a higher energy state, setting the stage for spontaneous emission. Second, his theory predicted that as light passes through a substance, it could stimulate the emission of more light.
Einstein postulated that photons prefer to travel together in the same state. If one has a large collection of atoms containing a great deal of excess energy, they will be ready to emit a photon randomly. However, if a stray photon of the correct wavelength passes by (or, in the case of a laser, is fired at an atom already in an excited state), its presence will stimulate the atoms to release their photons early–and those photons will travel in the same direction with the identical frequency and phase as the original stray photon. A cascading effect ensues: as the crowd of identical photons moves through the rest of the atoms, ever more photons will be emitted from their atoms to join them

You ask: "How did Einstein know that it was necessary to include "stimulated emission" which was in the same direction as the incident photon and in phase with it? "
The simplest model , spontaneous emission, would radiate isotropically in angles and thus a wave could not be built up coherently. It is a  postulate of his model that  direction and phase are the same " photons prefer to travel together in the same state".
The "preferred" in this model is a guess that became a mathematical certainty with the theory of quantum mechanics and its probabilistic predictions." Preferred" in quantum mechanics means "high probability" and comes out of the mathematical formulation.
A: It seems this was asked about 5 years ago, but I'll still answer to this. (Because I just can't take those show-offing comments who can't even answer to the question, but want to keep talking about random things. And this is my very first comment on the stackexchange)
Before Einstein, people at that time could only think of "Stimulated absorption" and "Spontaneous emission". Stimulated absorption is obviously a known process (I mean, there exists no chemistry without it) so I don't need to talk about it. But also at that time, people at least had the knowledge of radioactive decay processes (Thanks to Marie Curie). So they could think of Spontaneous emissions as well.
Now, Einstein thought "If nature can stimulate absorption, why can't nature stimulate emission?". By the way, he probably didn't think about relating it with the Spontaneous emission at first. So he's just thinking about the stimulated absorption concept. He assumed stimulated emission is also possible, and started working out the math. He first thought about a thermal equilibrium case. His equation was telling that there is something missing (about this, check Youtube MIT lecture). He immediately remembered "Spontaneous emission". He included this into the thermal equilibrium equation, and it surprisingly matched with the Planck's blackbody radiation formula.
So two ideas of his which other people didn't think of.

*

*If nature allows stimulated absorption, why can't nature allow stimulated emission?

*Assuming it's possible, we may have to also bring "Spontaneous emission" into the equation to satisfy thermal equilibrium.

A: According to this article here:
https://www.ias.ac.in/article/fulltext/reso/006/06/0028-0042
"Einstein as Armchair Detective: The Case of Stimulated Radiation" by Vasant Natarajan, which is a very well written and interesting description of what Einstein did, Einstein had to postulate stimulated emission in order for his thermodynamic model to be able to achieve equilibrium.
So the short answer is that he added it to balance his equations.
