Question

What exactly are free variables? Why is that the variables associated with columns with no pivots are chosen to be free variables?

Answer 1

1. You can freely assign any values, in the domain, to the free variables and find a solution for the system of equations. The basic(pivot) variables cannot be freely chosen i.e. their values depend on the values assigned to the free variables.
2. It is not essential that the pivot columns correspond to the basic variables (the ones that are not the free variables). It is just convention to do so. For example, if you have a system with 4 variables and elimination produces 2 pivot columns, you can freely choose two variables. The values the remaining two variables will take on depend on the values your chosen free variables take on.

Example:
a+b+c+d=4
a+b+c-d=3

Let a,b be the free variables: choose a=1, b=2:
then c+d=4-3=1 and c-d=3-3=0 => c=d=1/2
You can easily replace a,b with c,d in the argument above to exchange the free variables with the 'pivot' variables.