What is an n-by-n matrix in a series-parallel circuit?
in a series-parallel circuit all resistances must be the same.


This is nonsense. Having equal value resistors might make it easier to calculate the equivalent resistance of the network, but having different values isn't a physical impossibility and it won't result in destruction of the universe.

In the case of n parallel strings of n equal value resistors in series, the equivalent resistance of the combination is equal to the resistance of one of the individual resistors, so that's a particularly easy case to calculate.


  this example is connected in an n by n matrix. What exactly is this?


A matrix is a tabular arrangement of numbers, like the arrangement of resistors in your example. An n x n matrix has an equal number of rows and columns.


  The book explains a practical aspect of using series-parallel circuits by saying that a resistive component can be obtained with a large power handling capacity if there is a series-parallel configuration of resistors of the same value(as shown above in the diagram). Is there an explanation for this phenomena?


Say you put a current I through this (composite) resistor. 

The power consumed by the resistor will be \$I^2R\$. If you do this with a single resistor, the power is all consumed by that one resistor. If you do it with the n x n combination, each resistor only consumes \$\frac{I^2R}{n^2}\$. So the composite resistor can handle more power before any individual resistor fails due to overheating. 


  It says that this configuration can dissipate more power than just using a single resistor. What is power dissipation?


Power dissipation means converting power into heat. Since excess heat tends to damage things, resistors are usually rated for the maximum amount of power they can dissipate.