Sounds like exercise induced asthma, and I'll reiterate what Emily and Bikechick said - go see your doctor.
As to why the salt solution helped, probably for the same reason it helps when you have a cold or sore throat - the salt solution sooths the inflamed passages. But the WHY they're inflamed is the concern, and thus the reason for the trip to your local Doc.
Once you're on the right inhaler, you'll probably find you're riding better ... clear breathing, what a concept.

Good Luck!
Beth