Basically, Desktop Environments offer a complete desktop experience. The sort of 'normal' desktop experience most people have come to expect, from OSes like Windows, Mac, and most GNU/Linux distros. They're essentially a set of lots of different programs made by the same people, with the purpose of integrating together nicely. These programs tipically include: a window manager, a status/task bar, a nice login screen, a file manager, lots of little graphical programs for general daily usage, etc. People who say they're using 'stand-alone' window managers are essentially just making their own desktop environment, except they get to choose all the little programs that make up a desktop experience. So you choose what program will draw and manage your windows, what program will display your status bar, what program to manage your files, what terminal emulator, and so on.
It is important to note that the programs people use to make these set ups are usually very different from those offered by DEs. Usually DE programs are, well, made to be used with the DE. So all their configuration is handled by the DE's own 'settings app' or whatever, their features are made specifically to integrate with the DE, etc. The programs people usually use to make their own setups are a lot more standalone. They tend to use more universal interfaces rather than DE specific things, and are usually configured from their own configuration files, rather than some centralized UI. This allows for greater interchangeability: you don't have to change your window manager to change your status bar, for instance.
It's also worth a mention that DEs are not completely restricting either, unlike what I may have let on. Most DEs will allow you to use whatever terminal emulator you prefer as the default, and you can use whatever music player you want on any DE, etc etc. It's just that setting up a whole system from scratch usually yields in a more 'user-centric' experience, since the user gets to decide for themselves precisely which programs and settings to use to fit their uses and needs.
What makes these window manager setups, with standalone, interchangeable programs more desireable is precisely the fact that they're more barebones. They usually come with very simple configurations, and the user is left to figure things out. If you look at /g/ wiki's entry on GNU/Linux ricing, for example, you will see suggestions on all sorts of different programs to use, pic related. This is necessary, because all these features, which on a normal DE, come baked in, have to be set up from scratch.