The above text is taken from the book Molecular Dynamics Simulation: Elementary Methods by J. M Haile, Pub: 1992, Page-14.
Can you translate the equation (1.5) and (1.6) into plain English?
In 2D, integration means the area between a curve and the x-axis. In 3D, it would be volume. So, how can I visualize that?
Just to reiterate what we discussed in the comments: Since the position of each particle is given by $\mathbb{r}_i \in \mathbb{R}^3$, you're essentially integrating over the entire configuration space. The "region" that you're integrating over is a $3N$-dimensional volume, but it's easiest to think of it as just $N$-copies of $\mathbb{R}^3$, each of which represents the possible positions that particle $i$ can occupy.