Step 1: Solubility principle.
Solubility of organic compounds in water depends on their ability to form hydrogen bonds with water molecules. A stronger interaction with water means greater solubility.
Step 2: Hydrogen bonding by butan-1-ol.
Butan-1-ol has an $-OH$ group. Oxygen has electronegativity 3.5, making the $O-H$ bond strongly polarised. It forms strong hydrogen bonds with water molecules ($O-H \cdots O$).
Step 3: Hydrogen bonding by butan-1-amine.
Butan-1-amine has an $-NH_2$ group. Nitrogen has electronegativity 3.0, making the $N-H$ bond less polar than $O-H$. Hydrogen bonds formed by $-NH_2$ with water are weaker.
Step 4: Conclusion.
Because the $O-H$ group forms stronger hydrogen bonds with water than the $N-H$ group, butan-1-ol is more soluble in water than butan-1-amine. Butan-1-ol is more soluble in water because oxygen (electronegativity 3.5) forms stronger hydrogen bonds with water than nitrogen (electronegativity 3.0) in butan-1-amine.