Polar, cylindrical and spherical. Watch the little box of volume shrink as it approaches the axis, and see where the extra factors come from.
Slide the first coordinate toward zero and watch the boxes collapse. That shrinking is the whole reason for the extra factor.
Slide each coordinate and watch its surface light up.
Every coordinate system is a promise that three numbers name a point. Holding one number fixed and letting the other two run sweeps out a coordinate surface: in cylindrical coordinates those are a cylinder, a half-plane and a flat plane; in spherical they become a sphere, a cone and the same half-plane. The point you're looking at is where all three meet.
Now the part that matters for integration. In ordinary xyz coordinates a little box of side Δx, Δy, Δz has volume Δx Δy Δz wherever you put it. In these systems it does not. Two of the coordinates are angles, and turning through a fixed angle sweeps a long way out near the rim and hardly any distance near the axis.
So the boxes are fat far out and thin near the middle, and the honest volume carries a correction: r dr dθ dz for cylindrical, ρ² sin φ dρ dφ dθ for spherical. The two bottom rows are that claim under test — the formula's prediction against the box's measured volume. Shrink the box and they converge, exactly as they do for the Jacobian, because that is precisely what these factors are.
Push the second angle past 180° in spherical and watch the point climb back up the other side rather than continuing down. That's why φ is normally kept in [0, π] — beyond it, the naming stops being unique.