The Language Assignment gives you the ability to select the Content Language(s) which need(s) to be active for your box to be displayed. It is obviously used on multi-language Joomla! websites which use Joomla's native multi-language functionalities.
In the table below you can see which options are available to setup the Language Assignment.
Click in the textfield so a selectbox pops up from which you can filter the languages that you'd like by typing on the fly.
Once you select one you can continue typing to find the next one and so forth.
Of course the most obvious use case for the Language Assignment is when you want to differentiate the boxes into separate languages. You can have different boxes for different languages so that your users don't have to translate themselves.
It is important to note, in case it hasn't been clear enough, that the Language in question is the one used by your Joomla! and not the Language of the visitor's browser.