お見舞い申し上げます。

The word 'mimau' (見舞う), from which 'omimai' is derived, has a history dating back over a thousand years in Japanese literature. Its original meaning was simply 'to visit' or 'to inquire after someone's health or well-being.' Over time, 'mimau' specifically came to refer to visiting someone who is ill, injured, or has experienced a disaster, and offering words of comfort, support, or condolences. The prefix 'o-' (お) is an honorific, added to show respect and politeness. The verb 'mōshiageru' (申し上げます) is the humble form of 'iu' (言う, to say) or 'suru' (する, to do), meaning 'to humbly say' or 'to humbly offer.' It emphasizes the speaker's respect towards the person they are addressing or the person they are expressing sympathy for. Therefore, 'O-mimai mōshiagemasu' literally translates to something like 'I humbly offer my visit/inquiry (of sympathy/concern).' It is a very polite and formal expression used in situations where one wants to convey deep sympathy, wishes for recovery, or condolences.