Wrong: Yes, the explanation is incorrect.
(The statement is correct in the special case where Rgain is absent and input stages have unity gain. See next paragraph).
More wrong: In addition, the use of the term "buffer" is very misleading. The term "buffer" usually implies that a stage has unity gain, whereas the input stages usually have gain. In the special case where Rgain is open circuit (ie not used) then the input stages are indeed unity gain buffers and R1a and R1b are superfluous (ie could be short circuited). In cases where Rgain is present the input stages have gain.
Referring to your diagram
Correct statements would be that
- the voltage at the left hand end of R2 upper = V1 and
- the voltage at the left hand end of R2 lower = V2.
All following circuit references will be to the diagram below.
<-- Small version - higher res version below
ie Due to the opamp providing negative feedback to drive the difference between its inputs to zero, (V1 = Vx) and (V2 = Vy)
A much better way of drawing the circuit - The amplifier can be much more easily understood intuitively if it is redrawn as shown below (and in smaller size version immediately above). This is done in some instances but usually the version in the original query is used. This is such a good way of showing it that it is strange that it is not used more often. Consider the diagram below.
The top of Rgain (Vx) is at voltage V1
The bottom of Rgain (Vy) is at voltage V2
The voltage across Rgain = Vx-Vy = (V1-V2) = Vin
Now the magic.
It can be seen 'by inspection' that the same current flows from Vw to Vx, then from Vx to Vy and then from Vy to Vz (or A1_out through R1a, then through Rgain and then through R1b to A2_out).
SO the voltages across R1a, Rgain & R1b must be proportional to their resistances (as they all carry the same current). So, for example, if R1 = 7 x Rgain it must have seven times the voltage across it that Rgain has.
BUT the voltage across Rgain = Vin = V1-V2 from above.
Use R1 for R1a or R1b as result is symmetrical.
So the voltage across R1 = Vin x R1/Rgain
So gain of 1st stage = (R1 + Rgain + R1 ) / Rgain
= (2xR1 + Rgain)/Rgain or = 2xR1/Rgain + 1
Which is the classic instrumentation amp gain expression. The gain of the input stage may be altered simply by altering Rgain.
The output stage is a standard differential amplifier with stage gain = R3/R2
Overall gain = (2 x R1/Rgain + 1) x R3/R2
The above explanation is mine but the marvellous redrawing of the standard circuit comes from Wikipedia's Instrumentation amplifier page. Look there for their explanation of the same circuit and a lot more.