"Goes to swimming" is an awkward English phrasing. If he is just swimming, you would say "goes swimming," but if he is going to swim lessons you could "goes to swimming lessons." I've never heard it said as "goes to swimming" though.
You're right: this may not be "queen's English", but this phrasing is a commonly-used English construct, at least in my part of the world (NZ), and as a literal translation, it makes learning the Danish easier for newcomers.
eeeh... first time I see "svømning" in the whole danish course. I thought there wasn't any -ing form in danish so I'm quite confused here. Can other verbs have that -ning ending ? what's the rule about that ?