sorry to trouble you, but ...; excuse me, but ...
A polite, formal phrase used to preface a request, question, or apology. Often heard in business settings, customer service, and formal letters. The が at the end softens the statement and implies a following request or explanation.
I'm sorry to trouble you, but could you tell me your name again?
恐れ入りますが、本日は定休日となっております。
I'm very sorry, but we are closed today.
恐れ入りますが is a more humble and formal way to preface a request or objection than 失礼ですが, often used toward customers or superiors.
From the humble verb 恐れ入る (to be sorry, to feel apologetic) in its polite -ます form, plus the conjunction が (but). The phrase literally means 'I am sorry, but...' and has become a fixed polite expression.