The Friedmann equation governs the dynamics of the universe.

From this equation the big bang is derived so it's nice to have an intuitive feel for the type of physics that gives rise to it. It comes out of Einstein's theory of general relativity but a classical derivation gets us most of the way there and gives some physical insight into what the equation represents.

We can consider the Friedmann equation as an equation of energy conservation.

Consider a test particle of unit mass on the surface of a sphere enclosing mass M. The potential energy of the particle is given by:

V = - GM / R(t)

The kinetic energy of the particle is given by

2T = (dR(t)/dt)2

From conservation of energy we know that

(d/dt)E = 0

Where E is the total energy of our system. If the system is conservative ( i.e. if energy is conserved)* then it can be prooven that the total energy is the sum of the kinetic enery T and the potential energy V. Luckly gravity is a conservative system so

E = T + V

so

(d/dt) { (dR(t)/dt)2 - 2GM/R(t) } = 0

integrate wrt to time and let the constant of integration = B for the moment. we then obtain the following equation:

(dR(t)/dt)2 - 2GM/R(t) = B

now the mass is just given by

M = 4 pi rho R(t)3 / 3

putting this in results in the following equation:

(dR(t)/dt)2 - 8G pi rho R(t)2 / 3 = B

which is the Friedmann equation. In order to obtain the constant B you need the full machinery of general relativity. It turns out that B = -kc 2 where k is the curvature index and c is the speed of light.

The essence of the equation is that it is an equation of energy conservation. That a Newtonian treatment should give an answer so close to the answer given by general relativity is a little surprising. The two theories diverge on small scales e.g. the perihelion advance of mercury and the deflection of light by the sun are different in the two theories. Although the derivation of the Friedmann equation gives an approximately similar answer without general relativity the Newtonian treatment would never have anything to say about curvature. This represents a fundamental difference in the two worldviews. I think the fact that the constants come out right on the left hand side of the equation might have something to do with the ubiquitousness of Gauss's theorem but that is rather a speculation on my part.

For reference i suggest any introductory text on cosmology. MV Berry "Principles of Cosmology and Gravitation" is very light and a nice technical introduction. John Peacock's Cosmological Physics gives some deep insight into these questions.
* an example of a system that is non conservative is any system with friction. This is why we have no perpetual motion devices on earth.