A device administrator, when talking about an app, is basically an app which had extra privileges over other non-administrator apps. These "administrator" privileges do not require root access and only give the app certain extra functionality.
An app with root access is different and on a much higher and more dangerous level system-control wise. An app with root permissions can essentially do anything to your phone - delete virtually any file, secretly route your traffic to somewhere else, access things it otherwise wouldn't have access to.
However, that's not really a fair justification of what rooting can give you as it also provides benefits too, however this isn't really related to the question.
Rooting a device (to gain root privileges and to give an app root access) requires a process which usually voids a device's warranty, whereas for an app to become an app administrator you just need to enable it in your settings. A similar procedure is required for apps to request root access. However, while a device administrator can be enabled by tapping a switch and confirming a dialog, root access is usually presented to the user as a popup dialog with a timeout.
Also, the (main and most probable) reason why your app locker app requests to become a device administrator is because device administrators cannot be uninstalled without you first disabling them (this process is basically the same as enabling it, however without the confirmation dialog). This is because someone who might want to access an app which you have locked might just uninstall the app locker to access it - this adds some security in that they have to spend the (albeit small) amount of time disabling it, and if they don't know how to (as this isn't something you do every day, even though it is simple) it helps prevent them gaining access to apps you don't want them to.