"I know what you did in July."
Translation:Jag vet vad du gjorde i juli.
att is commonly used where you would put the word "that" into a sentence. Example: I know that you did it in July ... Jag vet at du gjorde det i juli. In the above example, no "att" is necessary. That's my understanding of it anyway. It's easy for me, knowing Spanish, where the particle word "que" serves the same function as "att" here.
Because this would be to “know” in a sense of “connaître” in French or “kennen” in German. It means that you have met someone previously and therefore recognise him when meeting him again. You know the person's name, age, place of residence, etc. Something like that. “vet”, in this sentence, means that someone has knowledge of something, like someone else's activity in last July. In French, this would likely be “savoir”, and in German, it would be “wissen”.