My mother was busily cooking. Meanwhile, I was watching TV.

Literal

Mother [topic-は] busy-looking-[adverb-に] cooking was-doing. That while I [topic-は] TV [object-を] was-watching

忙しそうに = 'looking busy' — そう (appearance) + に (adverb). その間 = 'meanwhile/during that time.' Two parallel ていた clauses contrast the mother's activity with the speaker's.