Well, I believe that the first one means "on one Saturday" (probably the closest one), and the other "on Saturdays generally"...
It depends I think: I would take "On Saturday I am cleaning my room" to mean this coming Saturday only, but "On Saturday I clean my room" probably to be a weekly thing. Having said that though, I can see this might be one of those occasions where not accepting a particular right answer might help the learning process...
Days of the week are exceptions, they take Accusative even though you expect Locative. Also "weekend" behaves the same.
This time I read the Polish sentence before translating, so I got it right.
However, the audio sounded like "w sobotę sprzątamy mój pokój". After listening to it a few times, the beginning sounded like either "soboty" or "sobotę". But the middle word kept sounding like "sprzątamy" and not like "sprzątam".
"Saturdays I clean my room" should be accepted. The "On" is not strictly required.