Many people say this sentence sounds incomplete and I agree. However, I am a native German and we have exactly the same thing in German.
"Ich bin zu Besuch" is literally "I am to visit" with "visit" being a noun.
This structure can only be applied to the word "visit" in German, though I am not sure whether this is also the case in Norwegian.
In German it is more of a saying. Now that I think about it it sounds incomplete and does not make any sense to people learning the language, but it is still right. I hope I could help with some of the confusion.
One of the many examples why direct translation with prepositions does not always work and one of the reasons where more answers should be accepted to focus on the topic instead of missing things already learnt.
Although if you say "I'm on visit" would really not be a scholar way of saying the sentence correctly, I'd like to know from a Norwegian is besøket used in a similar sentence sometimes?