Learn Danish: Han drømmer om sin egen autocamper. - He dreams about his own camper van.

Because sin is a reflexive possessive pronoun that refers back to the subject of the clause, Han. Using sin egen means he is dreaming about his very own camper van. If you used hans egen, it would mean he is dreaming about another man's camper van.

Why is the emphatic possessive spelled egen in this sentence rather than eget or egne?

The word egen behaves like an adjective and must agree with the gender and number of the noun it modifies. Since autocamper is a singular, common-gender noun (en autocamper), we use the basic form egen. If the noun were neuter, it would be eget, and if it were plural, we would use egne.

Why is the preposition om used here?

In Danish, the verb drømme (to dream) is standardly followed by the preposition om when expressing what the dream is about. This directly translates to both 'dreaming of' and 'dreaming about' in English.

Is autocamper an English loanword, and how does it decline in Danish?

It is a pseudo-anglicism used in Danish. Despite its English appearance, it is a fully integrated common-gender noun. If he were talking about a specific one, it would be autocamperen (the camper van), and if he dreamed of several, it would be autocampere.