My father takes pride in the fact that I'm handsome.
Literal
Father [topic-は], I [subject-が] handsome [nominalizer-の] [object-を] pride [into-に] is-doing.
自慢にしている means 'takes pride in / makes something a point of pride.' にする implies deliberately treating something as a source of pride. Distinct from 自慢している ('boasts about'), which focuses on the act of boasting.