Translation:That is the gentleman from the restaurant.
Certain prepositions trigger a particular type of article case. To be fair, you haven't reached that point in the tree yet though; it's a little further down.
Here are the notes from that section:
DATIVE PREPOSITIONS Dative prepositions always trigger the dative case. Here they are: aus, außer, bei, gegenüber, mit, nach, seit, von, zu
ACCUSATIVE PREPOSITIONS Accusative prepositions always trigger the accusative case. Here they are: bis, durch, für, gegen, ohne, um
TWO-WAY PREPOSITIONS Two-way prepositions take the dative case or the accusative case depending on the context. If there's movement from one place to another, use the accusative case. If there's no movement or if there's movement within a certain place, use the dative case.
Here they are: an, auf, entlang, hinter, in, neben, über, unter, vor, zwischen
No movement -> dative: Ich bin in einem Haus (I am in a house)
Movement within a certain place -> dative: Ich laufe in einem Wald (I am running in [within] a forest)
Movement from one place to another -> accusative: Ich gehe in ein Haus (I am walking into a house)
In British English you most certainly do say, "that's the man FROM/OUT OF the restaurant". Herr in this case means "gentleman" rather than "sir". It would be like spotting someone in the street and saying " oh look. There's the guy from the restaurant/bank/ supermarket". Or if watching a film and you recognise and actor from another film. "That's the guy OUT OF Titanic". Aus is working exactly like English here.
Am I wrong to assume that "mister" would be the most appropriate translation of "Herr"? "Lord" and "master" and even to an extent "gentleman" seems rather archaic and outdated unless one lives in a feudal society.
Duo does not agree with me but proposes "master". Can't really see myself referring to anyone in this day and age as "master" (unless very specifically as a master craftsman which in German would correspond to "Meister").
Should I report it?
"Das" and "der Herr" (from the restaurant) are the same subject and therefore in the nominative case. If I say: "Der Herr sieht dem Herrn aus dem Restaurant sehr ähnlich" (resembles him), then the second "Herr" is not the same but both have something together, therefore the second is in the dative case.