Argument and motivation behind maps used in finding inrreducibles in an integral domain I present the solution to a past question that I didn't had time to understand even on a superficial level.

I am unable to understand the classical argument employed in the methods for finding irreducibles. I have stared at this for 5 whole hours. 
Here's what I know: 21=7.3=$(1+2\sqrt{-5})(1-2\sqrt{-5})$
The factors 7,3, $(1+2\sqrt{-5}) and (1-2\sqrt{-5})$ are all non-unit and clearly are non-zero.
They satisfy ONE condition for the definition for an element/ factor to be an irreducible. 
To ensure the factors are irreducibles, we need to ensure that there exists two elements in the Integral domain $\mathbb{Z}\left [ \sqrt{-5} \right ]$ such that a factor, say 3 = xy where x,y are elements in $\mathbb{Z}\left [ \sqrt{-5} \right ]$. It suffice for us to ensure that either x or y are units.
But in the solution, the argument assumes that x and y are non-units. Is this a proof by contradiction?
Also, why it it necessary for a map N commonly used in determining primes, irreducibles and associates to exists?
I'm extremely confused when authors toggle between the image of the elements under the map N and the elements itself in the integral domain. 

Suppose that the map $N$ is $\rm\color{#c00}{multiplicative}$ $\,N(xy) = (Nx)(Ny),\,$ and, furthermore, that  $\,\color{#0a0}{x\mid Nx}.\,$ Then $\,Na$ irreducible $\,\Rightarrow\, a$ irreducible, since
$\ \ a = bc\,\color{#c00}\Rightarrow\, Na = Nb\, Nc,\ $ so $\,Nb\,$ or $\,Nc\,$ is a unit, say $Nc.\ $ Then $\,\color{#0a0}{c\mid Nc}\mid 1\,$ so $\,c\,$ is unit. 
Yours is special case $\,Na = a\bar a =  {\rm Norm}(a)$
Remark $ $ The key idea is that multiplicative maps preserve multiplicative properties (which is brought to the fore in divisor theory). Above we pullback the "atom" (irreducible) property  along the multiplicative norm map $\,N$. 
In fact much of the multiplicative structure of a number ring is reflected in its monoid of norms. For example, in many favorable contexts (e.g. Galois) a number ring enjoys unique factorization iff its monoid of norms does. See this answer for further discussion, including literature references (Bumby and Dade, Lettl, Coykendall).
