The standard reference for these sorts of facts is Silverman's book "The arithmetic of elliptic curves". The statement is that the $n$-torsion subgroup of $E(\overline{K})$,
which is naturally a $\mathbb Z/n\mathbb Z$-module (because it is an abelian group of
exponent $n$), is actually free of rank 2 over that ring (or, more concretely, it is
isomorphic to the product of two cyclic groups of order $n$). In fact, if $K$ has
positive characteristic $p$ (which is the case you are interested in) one needs the
additional hypothesis that $p$ does not divide $n$; otherwise the statement is not
true. (This is discussed carefully in Silverman's book.)

What do you mean by "how they are generated"? Do you mean to find explicit generators,
i.e. assuming that your elliptic curve has the form $y^2 = f(x)$ with $f(x)$ cubic
(as you may, at least when $p$ is odd), to find an explicit pair of points
$(x_1,y_1)$ and $(x_2,y_2)$ lying on the curve and defined over $\overline{\mathbb F}_q$
which generate the $n$-torsion subgroup (for some $n$)? If so, the classical way to
do this is by finding roots of the so-called division polynomials: these are polynomials
in $x$, whose coefficients can be written as (more and more complicated, the larger
$n$ is) expressions in the coefficients of $f$,
and whose roots are precisely the $x$-coordinates of the points of $E$ of exact order $n$.
(To find the corresponding $y$-coordinates one then just solves $y^2 = f(x)$.)

There are quite possibly better algorithms than this direct one, but I will let someone
with more expertise weigh in on that.

If you mean something else by "how are they generated?", then maybe you could explain more.

EDIT: I just saw your clarification. If $E[n] \subset E(\mathbb F_q)$, then $n$ will
necessarily be fairly small, since the order of $E(\mathbb F_q)$ is bounded above
by $1+ q + 2\sqrt{q}$ (the Hasse--Weil bound). In this case the division polynomials
will have some roots defined over $\mathbb F_q$, and you can find them explicitly given
enough computing power and the equation of $E$. Is this what you want?