Mother [topic-は] busy-[too-すぎて] I [subject-が] go-out [nominalizer-の] [to-に] noticed-not-[polite-past-でした]
忙しすぎて = 'too busy (and so...)' — ~すぎる (excess) in て-form links cause to result. 出て行くのに気付く = 'to notice the going-out' — のに marks what was (not) noticed.