Hi Kyle.
The short story is don’t use it directly unless you’re also going to plumb in all of the other bits and pieces that the calendar widget requires and uses. It’s documented purpose is, “Displays the mini grid calendar table (usually in a widget).” And it does, but it doesn’t do anything else.
If you want to use the calendar widget then the best course of action is to embed the actual widget itself inside a regular widgetized sidebar. By extension, if you don’t have a sidebar in the right place you would need to register a new one.
Trying to do anything else, whether using tribe_calendar_mini_grid() in isolation, trying to embed the widget using a call to the_widget() or some other method, is not currently recommended. We do have a bug logged and once it is addressed it should become possible to make use of the_widget() API function – however work on that bug is still to begin/still pending.
Sorry I can’t offer much else up on this occasion :-/