I am copying the setup from Sec 2.5 of these notes describing large deviations of an overlap matrix:
Let $\sigma_1,\cdots,\sigma_k\in\mathbb{R}^n$, with $k$ fixed and $n\rightarrow\infty$, and $\sigma_i\sim_{iid}\text{Unif}(\mathbb{S}^{n-1}(\sqrt n))$. We construct the matrix $\sigma=[\sigma_1,\cdots,\sigma_k]\in\mathbb{R}^{n\times k}$ and the overlap matrix $\overline{Q}(\sigma)=\frac 1 n\sigma^T\sigma\in\mathbb{R}^{k\times k}$. For some fixed symmetrix matrix $Q\in\mathbb{R}^{k\times k}$ with $Q_{ii}=1$ we are interested in: $$\lim_{\epsilon\rightarrow 0}\lim_{n\rightarrow\infty}\frac{1}{n}\log \mathbb{P}(\overline{Q}(\sigma)_{ij}\in[Q_{ij}-\epsilon,Q_{ij}+\epsilon],\forall i,j)$$
The notes then proceed (Eq. 17): $$\mathbb{P}(\overline{Q}(\sigma)\approx Q)\stackrel{.}{=}\frac{\int_{\mathbb{R}^{n\times k}}\delta\left(\overline{Q}(\sigma)-Q\right)d\sigma}{\int_{\mathbb{R}^{n\times k}}\delta\left(\prod_{i=1}^k||\sigma_i||^2/n-1\right)d\sigma}$$
The vague intuition is that the numerator is measuring the number of ways $\overline{Q}(\sigma)\approx Q$ and the denominator is a normalization over $\sigma_i\in\mathbb{S}^{n-1}(\sqrt n)$.
I guess that this is a way of rewriting an expectation over ($k$ copies of) $\text{Unif}(\mathbb{S}^{n-1}(\sqrt n))$ into an integral over $\mathbb{R}^{n\times k}$, i.e. it is rewriting the expectation: $$P(\overline{Q}(\sigma)\approx Q)=\mathbb{E}[\delta(\overline{Q}(\sigma)-Q)]$$
But how does one get the above ratio of integrals from this expectation?