You downloaded the wrong hardware architecture variant (e.g., x86 instead of ARM).
On your Android TV, go to Settings > Security & Restrictions and enable "Unknown Sources" to allow the installation of apps from outside the Google Play Store.
Search reputable APK archives (APKMirror is the industry standard for safety). Do not click random "Download Now" buttons on SEO-spam sites; look for the version history list for "YouTube for Android TV." youtube+for+android+tv+20702+download+link
Short summary
You can download this specific APK from reputable third-party archives: APKMirror - YouTube for Android TV 2.07.02 You downloaded the wrong hardware architecture variant (e
A conflicting version of YouTube is already active, or the file architecture does not match your TV processor (e.g., trying to install an arm64 build on a 32-bit armeabi TV).
Download the YouTube for Android TV 2.07.02 file from the link above onto a computer or directly to your TV. Do not click random "Download Now" buttons on
Download the exact version 20.7.02 package onto your phone or computer, then send it to your TV over your local Wi-Fi network.
: Features the signature clean, remote-friendly grid layout built for standard D-pads and physical navigation.
What of Smart TV or streaming box are you using?