Origins
Something proper originally belonged to you. The word comes from Latin proprius — 'one's own, particular, special'. A proper noun is a name that belongs to one specific thing. Property is what is properly yours. Propriety is behaving as your station requires.
The journey from 'own' to 'correct' followed a logical path. What belongs to someone is genuine. What is genuine is authentic. What is authentic is fitting. What is fitting is correct. Each step was small; the total distance is enormous.
Old French propre inherited all these senses and added another: clean. Something proper to a well-kept person is cleanliness. French still uses propre for both 'own' (mon propre fils — 'my own son') and 'clean' (une chambre propre — 'a clean room').
Middle English
Middle English pushed the word further. Chaucer used proper to mean handsome — a 'proper man' was a good-looking one. This sense survived into early modern English before fading.
The modern British colloquial use — 'a proper mess', 'proper angry' — works as an intensifier meaning 'truly, genuinely'. This is not slang degrading a formal word; it is the oldest meaning resurfacing. Proper has always meant 'truly, in the real sense'. A proper mess is a genuine mess — one that properly belongs to the category.