"Faren forklarer, at han er i live."
Translation:The father explains that he is alive.
Normally, I'd agree, but in this context that sounds a bit weird to me. However, I fully agree with that when it comes to sentences like "I know you're lying to me"/"I know that you're lying to me" and "He believes he can fly"/"He believes that he can fly". "He explains he is alive"… I don't know. Maybe they just don't talk like that in my region.
What is definitely wrong is the comma before "that" - not necessary in English.
Not in Danish. That is the correct way to write this sentence in Danish; without the comma, it would be wrong and sound weird to a Dane.
The English sentence did have a comma before, but I have removed it when I saw the comment yesterday. (Replying here as I can't reply to your most recent comment)