Why Is the DFT of a Signal Symmetric About Its Central Bin?
First of all it is only conjugate symmetric if the input signal is Real Signal.  

The reason for that is the DFT is built by Complex Exponential which can be decomposed into Sine and Cosine.
Since the Sine governs the Complex part and is $ N $ periodic one could see it creates conjugate symmetry of the DFT.

You just need to follow the definition of the DFT.