Bandwidth computation for NRZ baseband transmission scheme
The first statement says that for the given binary sequence of $$a_k = \{ 1,0,1,0,1,0,...\}$$ the transmitted NRZ line-signal would oscillate at the maximum rate : 
$$V = \{+A,-A,+A,-A,...\}.$$ which is alternating at a rate half of the binary data transmission bit-rate. This maximum frequency of line voltage oscillation should be capable of passing through the channel filter. Hence, the channel bandwidth should be at least 500 Hz in that case.

For example, if you want to transmit binary data at 1000 bits per second, by using NRZ, then the line pulse voltage alternates maximum at 500 Hz fundamental frequency.

And the second statement then follows the first: for such an alternating signal, your channel bandwidth should be at least 500 Hz (half the binary rate in Hz.) to a first approximation ideally. You should consider noise, pulse shaping and ISI issues for determining the required bandwidth more realistically.

Nyquist sampling theorem states that for a baseband signal of bandwidth, say, 4000 Hz, you need to take at least 8000 samples per second to exactly reconstruct the continuous signal back. And if you use 8 bits per sample, then that requires 8 x 8000 = 64 kbps binary data transmission rate. Note that such a binary rate would require a channel bandwidth of at least 32 kHz if NRZ is used.