Every nose has a natural pinch point called the nasal valve, roughly where the bony bridge of your nose gives way to soft cartilage. It's normally the narrowest part of your entire airway — narrower than your throat, narrower than your windpipe. A nasal strip's spring-like band, once applied, tries to return to its flat resting shape, and in doing so it gently lifts and holds the sides of the nostril open at exactly that point. Less pinch, less resistance, easier airflow — usually within seconds of applying it, and lasting up to around 12 hours per strip.
That's the entire mechanism. It's not absorbed, it doesn't interact with mucus or allergies chemically, and it does nothing once you take it off. Purely external, mechanical, and reversible.