It's the correct equation of motion that describes free spin $1/2$ particles.
Layman
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
Student
For an elementary derivation and a discussion of what the solution of the Dirac equation mean, see Physics from Symmetry by Schwichtenberg
Alternative, two possible derivations can be found at page 100ff in Relativistic Quantum Mechanics by Paul Strange