You do have to see it. I insist.
The Germans are singing “Die Wacht am Rhein” (“The Watch on the Rhine”, an 1840 song about protecting the Fatherland from French invasions led by any more Napoleons), not the Nazis’ “Der Horst Wessel Lied”.
zoogirl has it nailed, but I’d go further. Its dramatic importance is from the situation - at that time, France had been defeated, half of it was occupied directly by the Germans, and the other half (which still held Morocco as a colony) was ruled by a Nazi-collaborationist government based in Vichy. Up until that point in the movie, the people in Casablanca, a transit port for refugees on the way to neutral America, had cooperated resignedly with the German overseers who controlled the Vichy government, represented by police Capitain Renaud. When they played the inflammatory anti-French song around the bar’s piano, though, that suddenly tipped and everyone showed by their singing of the even-more-inflammatory “Marseillaise” (check out the lyrics sometime if you don’t know it) that they didn’t accept Nazi rule, they still loved France, they were going to fight back as much as they could, and were determined to drive the invaders back out. The suddenness of the swing in apparent sentiment is the source of the scene’s drama.
The scene immediately following frames it well. Right after the patrons finish “La Marseillaise”, the German commander orders Capitain Renaud to close the bar to stop the incipient riot. When Rick asks Renaud why, his reply is “I am shocked, shocked to find there is gambling on the premises!” The dealer stuffs some money in Renaud’s pockets with the words “Your winnings, sir”.
I would have picked Kevin Costner diffidently, softly, not quite crying, asking his father’s ghost to play catch. Gets me every time.
For sheer drama, give me the re-entry scene - the communications blackout period lasting way too long, everyone on the ground gradually realizing that the capsule had burned up and the crew was dead, the classroom teacher gently placing his hand on Lovell’s son’s shoulder, and suddenly the radio reply!