RetroPie currently does not provide a pre-made image for the Raspberry Pi 5. Also, the latest image at the time of writing this guide is v4.8 based on Raspbian with Debian 10 from March 14, 2022.
The GitHub for RetroPie has active development and mostly supports the Raspberry Pi 5 and the latest Raspberry Pi OS with Debian 12. So a manual installation is possible, but the official documentation is outdated and covers a wider range of hardware, which can make it hard to follow.
This guide will cover performing a manual installation specific to Raspberry Pi 5 hardware and will attempt to install RetroPie the same way it would be installed from a pre-made image.
I also tried to make this guide as detailed as possible with tons of screenshots to prevent user error, but it's really not all that many steps if you are already familiar with how things work. So the steps below are a high level overview for advanced users, but each step contains a link to detailed steps for users new to Raspberry Pi's and/or RetroPie.