As I mentioned in comments, professor Jack Simons offers a relatively simple physical interpretation of the idea of mixing in excited detrminant for treating dynamical correlation in his book "Quantum Mechanics in Chemistry" (Chapter 8) freely available here, as well as in this video which is also authored by him.
An important mathematical finding is that a linear combination of a reference determinant and a doubly-exicted one can be expressed as linear combination of two other determinants. Namely,
\begin{multline}
c_1 | \dotsc \ \phi_a \alpha \ \phi_a \beta \ \dotsc |
-
c_2 | \dotsc \ \phi_r \alpha \ \phi_r \beta \ \dotsc |
= \\
\frac{c_1}{2}
\Big(
| \dotsc \ (\phi_a - c \phi_r) \alpha \ (\phi_a + c \phi_r) \beta \ \dotsc |
-
| \dotsc \ (\phi_a - c \phi_r) \beta \ (\phi_a + c \phi_r) \alpha \ \dotsc |
\Big) \, ,
\end{multline}
where $c = \sqrt{c_2/c_1}$.1
Here, the determinants to the left differ by a doubly occupied spatial orbital $\phi_a$ being replaced by a doubly occupied spatial orbital $\phi_r$, while the determinants to the right describe the singlet $(\phi_a - c \phi_r)^1 (\phi_a + c \phi_r)^1$ state. So, a state created by adding the doubly-excited $| \dotsc \ \phi_r \alpha \ \phi_r \beta \ \dotsc |$ determinant to the reference $| \dotsc \ \phi_a \alpha \ \phi_a \beta \ \dotsc |$ one is equivalent to a state in which one electron occupies $\phi_a - c \phi_r$ spatial orbital (being in any spin state) while another electron occupies $\phi_a + c \phi_r$ spatial orbital (also being in any spin state). And this is the way electrons "avoid" each other: by occupying these different spatial orbitals.
For example, $\pi^2 \rightarrow \pi^{*2}$ configuration mixing in alkenes or $\mathrm{2s^2} \rightarrow \mathrm{2p^2}$ configuration mixing in alkaline earth atoms produce left-right polarized and top-bottom polarized spatial orbital pairs shown below.

Here one electron stays closer to the left carbon atom by occupying $\pi^2 + c \pi^{*2}$ orbital, while another avoids it staying closer to the right carbon atom by occupying $\pi^2 - c \pi^{*2}$ orbital.

In this case one electron stays closer to the top of an atom by occupying $\mathrm{2s} + c \mathrm{2p}$ orbital, while another avoids it staying closer to the bottom of the atom by occupying $\mathrm{2s} - c \mathrm{2p}$ orbital.
1) Here is the proof for $2 \times 2$ determinants, where for brevity $\phi_i = \phi_i \alpha$ and $\phi_i^* = \phi_i \beta$.
\begin{align}
c_1 | \phi_a \ \phi_a^* | - c_2 | \phi_r \ \phi_r^* |
&=
\frac{c_1}{2} \Big( 2 | \phi_a \ \phi_a^* | - 2 c^2 | \phi_r \ \phi_r^* | \Big) \\
&=
\frac{c_1}{2} \Big( 2 (\color{red}{\phi_a \phi_a^*} - \color{green}{\phi_a^* \phi_a}) - 2 (\color{blue}{c^2 \phi_r \phi_r^*} - \color{purple}{c^2 \phi_r^* \phi_r}) \Big) \\
&=
\frac{c_1}{2} \Big( 2 (\color{red}{\phi_a \phi_a^*} - \color{blue}{c^2 \phi_r \phi_r^*}) - 2 (\color{green}{\phi_a^* \phi_a} - \color{purple}{c^2 \phi_r^* \phi_r}) \Big) \\
&=
\frac{c_1}{2}
\Big(
(\color{red}{\phi_a \phi_a^*} - \color{blue}{c^2 \phi_r \phi_r^*})
-
(\color{green}{\phi_a^* \phi_a} - \color{purple}{c^2 \phi_r^* \phi_r}) \\
&\phantom{=\frac{c_1}{2}}-
(\color{green}{\phi_a^* \phi_a} - \color{purple}{c^2 \phi_r^* \phi_r})
+
(\color{red}{\phi_a \phi_a^*} - \color{blue}{c^2 \phi_r \phi_r^*})
\Big) \\
&=
\frac{c_1}{2}
\Big(
(\color{red}{\phi_a \phi_a^*} + c \phi_a \phi_r^* - c \phi_r \phi_a^* - \color{blue}{c^2 \phi_r \phi_r^*})
-
(\color{green}{\phi_a^* \phi_a} - c \phi_a^* \phi_r + c \phi_r^* \phi_a - \color{purple}{c^2 \phi_r^* \phi_r}) \\
&\phantom{=\frac{c_1}{2}}-
(\color{green}{\phi_a^* \phi_a} + c \phi_a^* \phi_r - c \phi_r^* \phi_a - \color{purple}{c^2 \phi_r^* \phi_r})
+
(\color{red}{\phi_a \phi_a^*} - c \phi_a \phi_r^* + c \phi_r \phi_a^* - \color{blue}{c^2 \phi_r \phi_r^*})
\Big) \\
&=
\frac{c_1}{2}
\Big(
(\phi_a - c \phi_r) (\phi_a^* + c \phi_r^*)
-
(\phi_a^* + c \phi_r^*) (\phi_a - c \phi_r) \\
&\phantom{=\frac{c_1}{2}}-
(\phi_a^* - c \phi_r^*) (\phi_a + c \phi_r)
+
(\phi_a + c \phi_r) (\phi_a^* - c \phi_r^* - c^2 \phi_r \phi_r^*)
\Big) \\
&=
\frac{c_1}{2}
\Big(
| (\phi_a - c \phi_r) \ (\phi_a^* + c \phi_r^*) |
-
| (\phi_a^* - c \phi_r^*) \ (\phi_a + c \phi_r) |
\Big) \, .
\end{align}