If I understand correctly, this contruction is used to emphasize that it is right now that I am eating. In contrast, in English, we use the progressive "ing" form of verb constantly, even when there is no implied emphasis, where most languages would simply use a regular present tense.
It is a common informal American construction to say "I am sitting and eating," or "sitting and reading," as a way to indicate a progressive action. Similarly, you can say you're "standing and listening." I'm sure this construction has the same roots in both languages.
English have a time more then danish have. In danish we don't have the time, the -ingform represents (i hope this makes sense) so you cant really translate it directly. We Danes have a habit of putting -ing bminto every english sentence because it sounds more english to us So: Jeg sidder og spiser I sit and eat I am sitting and eating It means the same in danish
Translated literally, yes it does mean that. But as other people have said it is also used to emphasise that you are in the middle of carrying out a particular action right now. Check out notes for the lesson on the website if can.