She thought of that autumn day when she had run home from school with good news. She had been chosen for the relay team.
Literal
She [topic-は] good news [object-を] carrying school [from-から] home [to-に] running returned that autumn [possessive-の] day [possessive-の] thing [object-を] thought. She [topic-は] relay-team [possessive-の] athlete [to-に] was-chosen [explanatory-のだった].
A two-sentence narrative passage. The first sentence contains a complex relative clause modifying 日 ('day') — everything before あの秋の日 describes what happened on that day. のだった in the second sentence is the past explanatory form, providing background context like a narrator's aside: 'the thing was, she had been selected.' とんで帰った (飛んで帰った) is a compound meaning 'flew/ran home' — figurative.