newtonian mechanics - What is the meaning of the Equation of Geodesic Deviation?

I've seen the Equation of Geodesic Deviation stated in several text as:$$\frac{D\xi^2}{dt^2}+Riemann(\textbf U,\xi,\textbf U)=0$$ but I haven't seen a real good explanation of why it works or the meaning of the right hand side of the equation. Why is it zero? Some texts and posts seem to suggest that this is Newton's First Law at work or an assumption of Minkowski space (objects at rest stay at rest, objects in motion stay in motion, acceleration is zero). Is the zero on the right hand side of the equation an expression of Newton's First Law of Motion?

EDIT: The Equation of Geodesic Deviation appears to be asking the question, what shape would allow these objects that appear to be accelerating (e.g. the Moon, the Earth, the Sun), to be in "free fall". The Riemann appears to be the answer to the 'what shape' part of the question, but the right hand side, zero in Newtonian Dynamics, appears to be the desired answer of the equation: a test particle in "free fall" with respect to a fudicial geodesic.