They may not have worded it well, but the usual approach to this analysis is to look at the case for v_A = v_B (that is, the AC signal component of each, not counting DC bias), and for v_A = -v_B.
In the former case, we get a wholly common mode input, and R_I is relevant: both outputs have a signal R_L / (R_I + 1/gm), in phase (same sign, and opposite the input sign). Note that, because there's no left/right difference, we can simplify the circuit by merging the sources and transistors, and now it looks like an ordinary common-source amplifier, albeit with an unusually large source resistance, so the gain is low (but finite).
In the latter case, we get a wholly differential input, and the source voltage hardly changes at all, so R_I can be canceled out. Gain is then simply gm * R_L.
Finally, because this is a linear circuit (in the small-signal case), we don't need to analyze these cases separately -- we can apply inputs v_A = v_CM + v_D and v_B = v_CM - v_D, and the outputs will similarly be the sum of CM and D cases. That is, the amplifier satisfies the condition,
For Vout as a function of Vin,
Vout(a(b + c)) = Vout(a) Vout(b) + Vout(a) Vout(c)
In other words, all the usual friendly associativity-distributivity-etc. properties we like numbers to have.
These are not true in general, for large signals -- nonlinearities violate these in various ways. The small-signal analysis assumes linearity, so it's a very handy tool when it can be used.
Tim