This is the 11492nd most frequent Russian word.
домашним
"Домашним" means "domestic" or "related to home."
The word 'домашним' relates to 'заданием' (assignment) here, meaning 'homework assignment', used in the instrumental case.
Мне нравится помогать сестре с домашним заданием.
I enjoy helping my sister with her homework.
The word 'домашним' modifies 'делом' (work), indicating activities related to home or household.
После долгого дня я предпочитаю заниматься домашним делом.
After a long day, I prefer doing household work.
The word 'домашним' describes 'гарниром' (garnish), implying it was made at home.