What is meaning of 2n factors? Can any one explain what does 2n factors mean and how it's been written n factors in next step. I tried to figure out but couldn't make out.
It means that there are 2n terms, each term being a factor of the form $(1+\omega^2)$. In the second step, they've grouped two terms together, for eg. $[(1+\omega)(1+\omega^2)]$ is one term and so, there are n such factors.
It means number of our factors and there are even number of factors like 8 factors or 10 factors whatever
let me solve it to understand your answer why and how we final get the result= 1
your question prove that: $(1+ω)(1+ω^2)(1+ω^4)$ etc.......to 2n factors $=1$
so they are factors multiply each other and they equal to 1
how we multiply  "product" factors
let me show you how we multiply factors 
now let's take like number 12 what is factors of "12"
$1\times12$ ,$2\times6$ , $3\times4$
so factor will equal ={1,2,3,4,6,12}
then how we multiply factors ? by the awesome formula
$N^{no.of factors/2}$
n=12 the number repeat by product of two numbers 
{$1\times12$ , $2\times6$ , $3\times4$}
number of factors{1,2,3,4,6,12} =6
by using the formula $12^{6/2} =1728$
Getting back on our question
note $ω^3=1$ , $ω+ω^2+1=0$
in the third bracket we will find 
$(1+ω^4)=(1+ω\timesω^3)=(1+ω)$
the first two brackets are repeat$=n$
and number of our factors $=2n$
let's use our formula
${((1+ω)(1+ω^2))}^{2n/2}$.two will go with two
${((1+ω)(1+ω^2))}^{n}$.multiply two bracket.${(1+ω^2+ω+ω^3)}^{n}={(1+ω^2+ω+1)}^{n}={(1+0)}^{n}=1^n=1$

i think that help you to understand the first answer .$\color{gold}{Yahia}$ $\color{gold}{Kamal}$ $\color{red}{♥♥}$ 
refrences:
https://www.youtube.com/watch?v=g4zTQveQuuY