> How can I complete the final step? I cannot build any connection between (2) and (3).

You need to use the completeness of the eigenfunctions

$$
\sum_i \psi_i(\vec r_1)\psi_i^*(\vec r_2) = \delta(\vec r_1 - \vec r_2)
$$

After plugging in the above completeness relationship on the RHS and your definition of the Green's function on the LHS, the resulting equation is:

$$
\sum_i \psi_i^*(\vec r_2)\frac{1}{(\omega - E_i)}\left \{(-\omega - h(\vec r_1))\psi_i(\vec r_1) - \int d^3r_3 \Sigma(\vec r_1,\vec r_3)\psi_i(\vec r_3)\right\}
=\sum_i \psi_i(\vec r_1)\psi_i^*(\vec r_2)
$$

The coefficients of $\psi_i^*(\vec r_2)$ have to be equal by completeness so:
$$
\frac{1}{(\omega - E_i)}\left \{(-\omega - h(\vec r_1))\psi_i(\vec r_1) - \int d^3r_3 \Sigma(\vec r_1,\vec r_3)\psi_i(\vec r_3)\right\}
=\psi_i(\vec r_1)
$$
Or
$$
(-\omega - h(\vec r_1))\psi_i(\vec r_1) - \int d^3r_3 \Sigma(\vec r_1,\vec r_3)\psi_i(\vec r_3)
=\psi_i(\vec r_1)(\omega - E_i)
$$

Cancel the $\omega \psi_i(\vec r_1)$ from both sides and then multiply both sides by $-1$ to get:
$$
h(\vec r_1)\psi_i(\vec r_1) + \int d^3r_3 \Sigma(\vec r_1,\vec r_3)\psi_i(\vec r_3)
=\psi_i(\vec r_1)E_i
$$