Father [topic-は], I [subject-が] tall [and-くて] handsome [nominalizer-の] [object-を] is-boasting.
A double-subject construction: 私が背が高い — 'I, as for height, am tall.' The outer が marks the person, and 背が marks the attribute. The whole clause is nominalized with の and marked as the object of 自慢する.