expression
goodbye (when leaving home); I'm off; see you later
Polite phrase said when leaving one's home or office, often to family or colleagues. Implies the speaker will return. More formal than 行ってきます.
See also: 行ってきます
「行ってまいります」と言って、彼は家を出た。
Saying 'I'm off,' he left the house.
毎朝、妻に「行ってまいります」と声をかける。
Every morning, I say 'I'm off' to my wife.
行ってきます is the standard casual/polite phrase for 'I'm off,' while 行ってまいります is more formal and humble, using 参ります (humble form of 行く).
Phrase composed of 行って (te-form of 行く, 'to go') + まいります (humble form of 参る, 'to go/come'). The humble 参ります adds politeness and implies a return.