It means you want to sum up infinitesimal volume elements ##dV##. A volume like a sphere has three dimensions, so three integrations are necessary. In cartesian coordinates you get the volume of a cube at the origin with size ##a## from $$
\int\limits_0^a \int\limits_0^a\int\limits_0^a dxdydz = a^3$$For a sphere the limits are unwieldy in cartesian, but comfortable in spherical coordinates. A volume element is ##r^2drd\phi d\theta## (##\theta## is azimuthal) so you get $$
\int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^R \; r^2 \sin\theta \;d\phi d\theta dr = \\
2\pi \int\limits_0^\pi \int\limits_0^R \; r^2 \sin\theta \;d\theta dr =
4\pi \int\limits_0^R \; r^2 \; dr = \ ...
$$
(There are alternative notations, like ##\ \int\limits_{\rm Volume} d^3 V\ ##)
