。
Mother is always bustling around.
Literal
Mother [topic-は] always busily move-around is-doing.
忙しく is the adverbial (連用形) of the い-adjective 忙しい — い-adjectives become adverbs by changing い to く. 動きまわる is a compound verb: 動く (move) + 回る (turn/go around), which fuses into 'move about, bustle around.' The whole thing sits under ~ている for the ongoing-action reading.