Translation:The old doctor sits on the balcony and reads.
It is implied by the -re ending. By itself, the English 'sits on the balcony' could be describing a static situation (he is already sitting) or an active motion (he sits down). The Hungarian sentence, using -re, makes it clear that he is actively taking a sitting position, then reading.
To be fair, he might be already standing on the balcony, and then decides to sit. It is not the DL authors who use the phrase 'going out', but richardkiss in his post above. But his interpretation is not unreasonable. In order to sit down on the balcony, you first have to go out to it.
Where did we learn that the -re ending indicates motion toward? I don't know whether it was in the Tips and Notes, or whether it was in disussion pages like this one.
In any case, yes, the -re ending is what is called sublative. It indicates that motion toward is involved, rather than a static situation.