You seem to understand the idea behind the
のだ format quite well. The difference I think you are seeking is between
The first is much more close and personal is used to give an explanation of some sort (this one is also harder to translate directly but I think your description of
the thing is is very close).
The latter is a much more literal, informational, and impersonal
It is that....
The first one sounds like an explanation provided the explains the speaker's situation. In this case it sounds like their friend is having trouble because he/she does not speak Japanese and is seeking understanding.
The second one sounds more impersonal and to the point. The speaker may have nothing at stake with the situation, like they are describing it as an observer.
Both, however, serve to give some reasoning and extra information and can be easily translated as
The thing is... or
It is that.... The question is the feeling behind it.