>>11624574f_g is the acceleration due to gravity at a point of evaluation. It's Newton's law of gravity, but applied to objects of finite density rather than point masses.
rho dV roughly represents the infinitesimal mass at some point in the domain of integration, little r is the distance between the point of evaluation and the infinitesimal mass, and r hat is the unit vector in the direction from one point to the other.
By integrating over the whole volume, you're adding up the contribution to the total acceleration from every little bit of mass in an object of finite density.
