Sampling with an alternating impulse train
We are not simply sampling the message in this case but also multiplying by a sampled tone that exists at $f_s/2$, (where $f_s$ is the sampling rate) and therefore we are frequency shifting the signal to this new location. 



This is intuitively clear from the two plots above. In the first case, representative of a sampling process, we are multiplying our waveform by an impulse train in time. Such an impulse train in time is a impulse train in frequency, with each impulse separated by 1/T where T is the time spacing between the impulses in time. Multiplication in time is convolution in frequency, so the process replicates the spectrum of the waveform at each of the impulses shown (as you showed you understood from the sampling process). Given the impulse that exists at frequency location 0, there will be a copy of the waveform spectrum here that is unchanged from the original, assuming no aliasing occurs from the other replicas at the other impulse locations in frequency (convolving with an impulse at freq = 0 does nothing).

In the lower plot we are doing something similar, but in addition we are multiplying the waveform by a alternating +/-1 sequence. Note that such a sequence in time has no DC value! The plot is very clear if you consider the time domain waveform to be a sampled cosine waveform, with a frequency that is exactly half the sampling period T. So in this case the waveform spectrum will be shifted away from the DC location; to each of the impulses as shown. 

As the OP has found, see this link that further details the mathematics: Expression to represent alternating pulse train?