Who is allowed to enter a street sign-posted like this?

This traffic sign means "No entry except for residents or people with a direct reason for access" (“Anlieger frei”). Only residents and visitors who need to reach a property or destination inside that street are allowed to enter. Through-traffic, meaning anyone just using the street as a shortcut without a direct purpose, is not allowed. You answered correctly with both residents and people visiting residents. This is important to keep local traffic minimal and prevent unnecessary transit traffic. Remember: signs with "Anlieger frei" always allow entry only for direct access, not for passing through.