"Beide" is an adjective.
"Beiden" is a pronoun (hence why it has "both of them" as a hint).
In fact, plural (pro)nouns that are derived from adjectives always get the ending -en. For example:
- Sommigen - Some people
- De sterken en de zwakken - The strong and the weak
- De anderen - The others
This lesson is showing you all the possible uses for demonstrative pronouns. One of the uses is to be next to a noun and become an adjective. I'm not sure about your mother tongue but my language has the same principle. Would you rather only be taught how to say "I see the others." , or "I see the other CHILDREN" as well? (especially since the demonstratives change form slightly when used as adjectivea-andere vs anderen). Hope that answers your question.
Dutch is quite lenient when it comes to tenses, so both Beiden zwemmen and Beiden zijn aan het zwemmen can be used to indicate this is currently happening (the latter has more emphasis that it is happening now). If it's about the habit of swimming, only Beiden zwemmen can be used.