Why DC potential at output of diiferential amplifier limits output voltage swing (peak to peak)?
What you read was probably talking about an amplifier circuit with a single-ended supply and directly-coupled output. Such circuits can't produce outputs outside the supply voltage range. So if the positive supply is \$V_{ss}\$ and the negative supply is \$V_{ee}\$ and the dc voltage of the output is \$V_{odc}\$, then the maximum output amplitude is 

$$\mathrm{min}\left\{V_{ss}-V_{odc}, V_{odc}-V_{ee}\right\}$$ 

And for if the amplfiier does not have rail-to-rail output capability, the maximum amplitude will be even smaller.

Then in order to maximize the available you should choose 

$$V_{odc}=\frac{V_{ss}-V_{ee}}{2}$$

with some slight adjustment if the outputs aren't rail-to-rail and the required margins are not equal on either side.