"We kunnen het antwoord vinden door na te denken."
Translation:We can find the answer by thinking.
I don't see the logic in the last four words together here. Is it just a phrase, or is there some implication beyond "by thinking", like maybe the "after" implies thinking in retrospect?
nadenken is the verb, split by "te". It means to reflect, consider, thing through.
Is it also correct to say nadenken instead of na te denken? (Like it would be in German ;) )
Nope. The construction is "door te" + infinitive. However, separable verbs always get split by "te".