He made up a story about having to go home to visit his sick mother.

Literal

Sick [possessive-の] mother [object-を] visiting [for-ために] must-return-home [quotative-といった] story [object-を] fabricated.

でっちあげる means 'to fabricate, to cook up (a story).' The fabricated story is quoted with ~といった before 話. 見舞う means 'to visit someone who is sick.' 帰郷しなければならない uses the formal obligation pattern for 'must return to one's hometown.'