"I hope that you eat."
Translation:Ik hoop dat je eet.
That has nothing to do with the number of verbs in a sentence. E.g.
- "Ik hoop dat jij doet waar jij goed in bent.
Three verbs and all conjugated accordingly to each respective subject.
- "Wij hopen dat hij vertelt waar jij naar school gaat.
You are probably confused with a sentence like:
- "Hij gaat slapen or "Ik kan zwemmen", which do use the infinitive (just like English)
"Jou" is the form of "jij" that is used for direct or indirect objects. It is similar to the English "me" as opposed to "I".
In this sentence, you is the subject of the subclause "that you eat", since you are the one doing the eating. That's why you can't say "jou". It would be the same as saying in English: You hope that me eat, which is obviously wrong. Unless you're the Cookie Monster.
dat (or any subordinating conjunction) does not cause inversion, but it alters word order by sending all of the verbs to the end of the subordinated clause.
What causes inversion is starting a sentence with something other than the subject (be it an adverb, an adverbial phrase or a subordinated clause), because Dutch is a V2 language (which means that in main clauses the verb must always be the second element -note that I used the word element and not word-, at least as long as it's a statement and not a question, when the verb becomes the first element).
Hope this helps!