"Een hond rent achter een kat aan."
Translation:A dog is running after a cat.
That would be: "A dog is running behind a cat." It might be accidental, or they might be in a race. A dog that is running 'after' a cat tries to reach the cat so that things can get interesting. A dog that is running behind a cat may be hunting elephants together with the cat and, feeling a bit uneasy about the next step, may gladly give its feline pal precedence.
I believe (but I'm not sure), that aanrennen is a verb that means "to run after" as in "to chase." Compound verbs like this split apart and the preposition at the beginning, "aan," gets relocated to the end of the clause. It's part of the Dutch grammar/word order. http://www.dutchgrammar.com/ is a good reference.
It's not proper English. The dog chases the cat or chases after the cat if it tries to reach the cat. It runs behind the cat if it doesn't try to reach the cat. It makes no sense for the dog to chase (i.e. try to reach) behind (i.e. without trying to reach) the cat.
If you run after someone, you are behind them.
You may also want to think of the achterdek = afterdeck of a boat as the behind of the boat.
In English, worse things have happened. The original meaning of fast is the same as Dutch vast: firm, fixed, hard, strong. Vestiges still exist, as in fast asleep. If you run fast in the original sense, you run rapidly. This is where the modern sense comes from.