My father is proud of being tall and handsome.

Literal

Father [topic-は] height [subject-が] tall-and handsome [nominalizer-のを] boasting.

自慢している (to be proud of/boast about) takes のを. 背が高くてハンサム chains an i-adjective (高い) with a na-adjective (ハンサム) via the te-form.