As in Weyl's treatment of compact Lie groups, Gelfand-Naimark for $GL_n(\mathbb C)$, Harish-Chandra's treatment of characters of reductive Lie groups $G$: the regular semi-simple elements $g$ form a set of full measure in the group, and the centralizer $Z(g)$ of regular semi-simple $g$ includes a maximal torus. For compact $G$, all these centralizers are conjugate, so $G/Z(g)$ is isomorphic as $G$-subspace (with conjugation) of full measure in $G$, and Weyl's character formula and dimension formula fall out. For complex reductive (following Gelfand-Naimark and Harish-Chandra) there's again a single conjugacy class. For real reductive, Hirai and Harish-Chandra had to worry about "patching conditions" for characters at the boundaries/interstices between the finitely-many conjugacy classes.
(A similar structural thing happens in p-adic reductive groups...)
Edit: in some further detail... with $A$ the diagonal subgroup in $G=U(n)$, for example, and $G'$ the full-measure subset of $G$ consisting of regular semi-simple elements, for a conjugation_invariant function $f$, $\int_G f = \int_{G'} f = \int_{G/A} \int_A f(gag^{-1}) = \int_A f(a) (\int_{G/A}1)$. A similar computation works, for example, in $GL_n(\mathbb C)$, or whenever there is a single conjugacy class, with the orbital integrals of general $f$ appearing. In Harish-Chandra's formulations of Plancherel in terms of characters, this integration formula is it, indeed. (I don't think it's really about Cartan decomposition or other of the standard decompositions...)