Why does menthol clear a stuffy nose?
Menthol is stimulating action potential at cells which are inducing thermoception. As far as I have found out it does not have any effect beyond that. The patients only feel like they can breath better while it could not be proven that they actual do.

https://www.ncbi.nlm.nih.gov/m/pubmed/18702906/

The blood vessel thing : In general "cold" is a cause for vasoconstriction - that might lead to the conclusion that menthol is doing the same. I have no experental background on that but - after online research :-D- it seems like that menthol causes vasodilatation.