Open at login is on by default: setup offers it at the last step, and you can change it in Settings › General › Open at login. It matters most on the Mac you use as a display, where nobody may sit after a restart.
From the Applications folder
macOS opens an app at login only from a place where it stays. If Second Display runs from the disk image or from Downloads, Settings says “Move Second Display to the Applications folder to open it at login.” Click Move to Applications…: Second Display copies itself there, quits and opens again from there.
If macOS asks for approval
Settings then says “Allow Second Display in Login Items in System Settings.” Turn it on there.
On macOS 12
On macOS 12 Monterey, add Second Display yourself in System Preferences › Users & Groups › Login Items. Settings has a button that opens that pane.