Translation:He swims every Wednesday and Saturday.
Definite time expressions use accusative case if there's no preposition, and dative case if there's a preposition. Indefinite time expressions use genitive case.
Accusative examples (no preposition):
Ich gehe dieses Wochenende ins Kino = "I’m going to the movies this weekend"
Wir lernen jeden Tag Deutsch = "We learn German every day"
Nächste Woche mache ich das = "I'll do that next week"
Ich hätte die ganze Nacht tanzen können = "I could have danced all night"
Wir haben ihn letzten Dienstag gesehen = "We saw him last Tuesday"
Dative examples (with preposition):
Ich fahre in einer Woche nach Chicago = "I’m going to Chicago in a week"
Am Montag esse ich oft Pizza = "On Monday I often eat pizza"
Am Wochenende fliege ich nach Paris = "On the weekend I'm flying to Paris"
Im Winter scheint die Sonne = "The sun shines in the winter"
In der Nacht sind alle Katzen grau = "At night all cats are gray"
Vor einem Jahr habe ich das nicht gewusst = "A year ago I didn't know that"
Genitive examples (indefinite time):
Eines Tages wird er alles verstehen = "One day he'll understand everything"
Eines Abends sollen wir ins Kino gehen = "We should go to the movies some evening"
Eines Nachts ist er aufgetaucht = "He showed up one night"
FYI, the rule about the prepositions = dative works because time expressions are usually not used as the goal of direction, so the rule is really just a reiteration of two-way prepositions taking the dative case when there is no movement. In poetry if you want to say something like "I ran into the night," the time expression behaves like any other noun and takes the accusative case. We're likely not to run into this issue in conversation or on Duolingo, though. I just thought it was interesting to point out.
Thanks for the outlining of the rules! It has been very helpful.
Think of 'each' as a shorthand for 'each one'. "Each Wednesday" would need to refer to an individual day, but in the example it refers to all the Wednesdays, not a particular one. Therefore one has to use "every" instead. We could say: There are seven days in the week. He swims on each one of them. because pick any one day, and he does go swimming. There is also the useful and common collocation each and every one, for example: There are seven days in the week. He swims on each and every one of them. Here "each" refers to each day individually, whilst every refers to all seven of them together.