How are the classical set of equilibrium equations for linear elasticity derived?

You take an arbitrary volume $V$ and use the translational and rotational equilibrium equations over it. Then, due to the arbitrariness of the volume the integrals should equal 0 and you get the equation you present and the symmetry for the stress tensor (in classic elasticity).

According to @BiswajitBanerjee's comment, the first publications to discuss the topic were:

• Navier, C. L. M. H. (1821). Sur les lois des mouvement des fluides, en ayant egard a l’adhesion des molecules. Ann. Chimie, 19, 244-260.

• Cauchy, A. L. B. (1822). Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques.

You can find a recent discussion on

• Mase, George Thomas, Ronald M. Smelser, y George E. Mase. 2010. Continuum mechanics for engineers. 3rd ed. Boca Raton: CRC Press. (Chapter 5).

• Reddy, J. N. (2013). An introduction to continuum mechanics. Cambridge university press. (Chapter 5).

You can find a discussion in AF Bower's book.

Applied Mechanics of Solids 1st Edition ISBN-13: 978-1439802472, ISBN-10: 1439802475

The book is available online at AF Bower's website

http://solidmechanics.org/Text/Chapter2_3/Chapter2_3.php#Section2_3_1

A different perspective on the question is this: Newton's law says that mass times acceleration equals the sum of all forces. You are interested in the steady state case, so the acceleration is zero and as a consequence, the sum of all forces is zero. This has to hold at each point of the solid if you want the body to not move.

The sum of all forces equals the external forces $F$ (actually, a force density, because we're looking at individual points) acting at each point of the body plus the internal forces $nabla cdot sigma$ due to the stresses.

In other words, the equation you quote is simply a force balance.