Etymologically, English infinitives (as well as Russian infinitives) are dative forms of nouns.
If you look long enough at the sentences "I give the dog to you" and "I teach the dog to fly", you will probably start seeing their similarity.
If you look long enough at the Russian sentences: я учу сестру плести and я приношу жертву нечисти, you will probably see their similarity as well.
Dative has long since died in English, but whenever you are saying "I teach to swim", in English and Russian alike, your are saying the phrase which has started out as "I teach to the skill of swimming", in dative.
Hope this will help you to wrap your head around it.