Suppose I have a graph, with a collection of nodes or vertices called V, and a set of edges between vertices such that if two nodes u and v are connected in the graph, then the pair (u,v) is in the set of edges, which I will call E.
A partial order is just an ordering of elements (think 1<=2<=3<=...) which doesn't have to specify the order between every object. Suppose that the topological order is partial.
First, because E is encoded as a collection of pairs of vertices, we say that E ranges over V*V. Therefore, we can do a bit of abuse of notation and say that "u E v" iff (u,v) is in E. That is, E is a "relation" on V.
Now, there's a natural order induced by the edge relation for any directed graph: if there's a path from u to v, then we can say that in some sense, u is "smaller" than v. This ordering is called the topological order of the vertices V.
The claim is that the ordering relation is the least fixed point over the composition of the edge relation E. That is, the relation we're seeking P is the smallest solution of the equation
where, P^2 stands for the composition between P and P, so if a P b and b P c, then a P^2 c.
In order to construct this P relation, we first make the observation that set V x V is finite, so its superset is finite and there are a finite number of possible relations "P", therefore any sequence of relations on V x V must be guaranteed to have a least upper bound (because it must be a finite sequence).
Next, suppose we construct progressively "better" approximations of the "P" relation, starting with the "bottom" element the identity, which says I don't know anything about where u can go to, except that it can reach itself. Then E^1 is the approximation of all of the nodes that u can reach in exactly one step, and E^0 \cup E^1 is the approximation of using 1 or fewer steps.
If we compose E with itself, then if u goes to w, and w goes to v, then u E^2 v. Therefore, E^2 is the relation from u to v if u can get to v in exactly two steps, and E^0 \cup E^1 \cup E^2 is the approximation of P accurate two two steps. Therefore, the union of {E^0, ..., E^k} is the approximation of P accurate to k steps. Let's call each of these approximations P_k
Finally, if we have a sequence of approximating relations that are becoming progresively closer to the true solution, such as
then it's guaranteed to have a least upper bound as k goes to \infty (because after a finite number of steps, it'll just go to P_n, P_n, P_n, ...), and that P_n is the convergence of the sequence of approximating relations (because each of the approximation is better/more informative than the previous, and we eventually converge) and also intuitively the least upper bound, that fully characterizes the relation P. We call this limit point the reflexive transitive closure of E, and we name it P_n = P = E^*. Proof of it being least follows from the tarski-knaster thm.
This is why the typical algorithm that computes the total topological order are framed as a computation of a least fixed point: we're looking for a least fixed point!
The take-away: topological sorting is the process of taking the limit of a sequence of approximations of an ordering relation.