Why did Einstein pick his definition of synchronous clocks? In his paper "On the Electrodynamics of Moving Bodies", Einstein defines synchronous clocks as clocks which satisfy $t_B-t_A=t'_A-t_B$. $A$ and $B$ are two points separated by empty space, and identical clocks are located at $A$ and $B$. A light signal is emitted from $A$ at time $t_A$ as measured by the clock at $A$. The signal arrives at $B$ at time $t_B$ as measured by the clock at $B$. The light signal is immediately reflected at $B$, and it returns to $A$ at time $t'_A$ as measured by the clock at $A$.
Why does he choose this definition for synchronous clocks? Why is this a sensible definition for synchronous clocks?
He does say this which might provide some insight into why he chose this definition: "We have not defined a common “time” for
A and B, for the latter cannot be defined at all unless we establish by definition
that the “time” required by light to travel from A to B equals the “time” it
requires to travel from B to A."
I can't seem to understand this quotation, because I'm not sure what he means by "time" in this sense. So far time has only be defined at the locations of the two clocks, so you can't use one clock to measure the time for the light signal to move between the points, you would have to use a measurement from both clocks, but there is not necessarily any relation between the time displayed by the two clocks as far as I understand the situation.
 A: Well, it is a sensible definition in the context of the completely new concept of space and time that special relativity introduced.
But you have to appreciate the problem that Einstein faced. How do you introduce something fundamentally new? How do you straddle a transition from an old theory of space and time to a fundamentally new one?
A few years later, in 1908, Herman Minkowski chose to state the fundamentally new insight in a very in-your-face manner:
"Henceforth space by itself, and time by itself, are doomed to fade
away into mere shadows, and only a kind of union of the two will
preserve an independent reality."
In his 1905 article Einstein didn't do that at all. As far as I am aware of historians of science don't know why Einstein chose to present his fundamentally new insight in a very implicit manner. But that is what he did. The form that Einstein chose was that he presented a rather abstract assumption: the speed of light is the same with respect to all the members of the equivalence class of inertial coordinate systems, and he proceeded to work out the logical consequences of that starting point.
The definition of synchronous clocks in Einstein 1905 is designed to dovetail with the logical implications of special relativity. Einstein needs that, otherwise he cannot proceed use it in the course of deriving those logical consequences. 
And yes, if you try to understand that definition in terms of the old concept of space and time (usually referred to as 'newtonian space and time') then the definition comes across as odd and awkward.
Again, how do you straddle a transition to a fundamentally new concept of space and time?
Once you have transitioned to the concept of Minkowski spacetime you recognize that the definition of synchronous clocks that Einstein gave is the proper way to define synchronous clocks.
A: 
Why does he choose this definition for synchronous clocks? Why is this a sensible definition for synchronous clocks?

The analysis of what is meant by time and simultaneity began before Einsteins paper of 1905. Poincare in the Measure of time published in 1898 in the Revue de Metaphysique et de Moral begins with:

So long as we do not go outside the domain of consciousness, the notion of time is relatively clear. Not only do we distinguish without difficulty present sensation from the remembrance of past sensations or the anticipation of future sensations, but we know perfectly well what we mean when we say that of two conscious phenomena which we remember, one was anterior to the other; or that, of two foreseen conscious phenomena, one will be anterior to the other.
When we say that two conscious facts are simultaneous, we mean that they profoundly interpenetrate, so that analysis can not separate them without mutilating them.
The order in which we arrange conscious phenomena does not admit of any arbitrariness. It is imposed upon us and of it we can change nothing.

However he goes on to say:

The trouble is that there is no rigor in the definition. When we use the pendulum to measure time, what postulate do we implicitly admit? It is that the duration of two identical phenomena is the same; or, if you prefer, that the same causes take the same time to produce the same effects.
And at first blush, this is a good definition of the equality of two durations. But take care. Is it impossible that experiment may some day contradict our postulate?

One of Einsteins achievements was to come up with a useful, rigorous definition of time.
A: This method of synchronization works because both clocks can agree on the speed of light.
If we want to determine whether or not two distant events are simultaneous, or more generally the order in which events occur, then we need to factor in the time it takes for the signals to reach us. We can easily find this time if we know the distance of the event, and the speed of the signal.
The constant speed of light  is convenient, since everyone can agree on its value, it's $c$. The method of synchronization also ensures distance is fixed, as the signal is returned immediately with no delay. So if both clocks can agree on the speed and distance, then it means they must agree on time (otherwise, they are out of sync).
