Step 1: Identify the structural difference between the two amines.
Propylamine ($CH_3CH_2CH_2NH_2$) is a primary amine with two N-H bonds. Trimethylamine ($(CH_3)_3N$) is a tertiary amine with zero N-H bonds; all three bonds on nitrogen are N-C bonds.
Step 2: Determine hydrogen bonding capacity.
Intermolecular hydrogen bonding in amines requires an N-H bond acting as a hydrogen bond donor. Propylamine can form $N$-$H \cdots N$ hydrogen bonds with adjacent molecules. Trimethylamine cannot form N-H hydrogen bonds with itself.
Step 3: Connect to boiling point.
The hydrogen bond network in liquid propylamine requires additional energy to break, raising its boiling point significantly above that of trimethylamine, which is held together only by weaker London dispersion forces.
Step 4: Assess assertion and reason.
The assertion (propylamine has a higher boiling point) is true. The reason (propylamine has intermolecular H-bonding while trimethylamine does not) is true and correctly explains the assertion.
\[ \boxed{\text{Both A and R are true; R is the correct explanation of A}} \]