"Læs" is a load (of something, say hay or stone). It is not a fixed amount but still "as much as can be transported/ loaded on a particular vehicle" - which in former times would be a horse-driven wagon or wheelbarrow and now more likely a truckload, but really anything. So the sentence means "how many (truck) loads (of some unspecified material) have you got?" Note that "læs" is also the imperative of "at læse", i.e. "read!" Quite many Danish words have two or even more meanings (try "for" or "far" or "får" or "leje" or "led"...). Context is important and usually the possible double-meanings are not much of a problem.
I could make sense in english with something as "laundry loads": https://es.answers.yahoo.com/question/index?qid=20140902224520AA8ONMl And if you Google "Hvor mange læs" you get many results. I see some related to truck loads or things like that. https://www.google.es/search?q=%22Hvor+mange+l%C3%A6s%22&oq=%22Hvor+mange+l%C3%A6s%22&aqs=chrome..69i57j0l5.1290j0j4&sourceid=chrome&ie=UTF-8
Brits tend to use "have got" more than Americans ("I've got three cats" in BrE vs. "I have three cats" in AmE). I don't know if it's grammatically incorrect, but it's certainly common enough that no one there would think twice about it. Even as an American I would say "The homework? Yeah, I've got it" as opposed to "I have it"/"I got it".
If you've seen or read any British movies/books (like Harry Potter) this construction is used a lot!
The audio is terrible. DUO: You have to discriminate between different use of a given word, i.e. the word (in Danich) læs. Here it should be pronouned differently from (to read a book .. læs en bog. The meaning here is different from this of couse in this sentence. You have to have a word bank with different pronunciations and you ought to have your audio rewied!