I know all the component parts to the sentence but can't see how they got to that translation. Literal translations aren't helpful but it would help me understand the use of the grammar if I could have a steer.
The translation of the expression may change depending on the situation.
おまたせいたしました (OMATASE ITASHIMASHITA) can be translated to “Thank you for waiting”, “I am sorry to have kept you waiting” or “We apologize for the delay.”
Waiters, shop assistants and office workers say the expression all the time.
TV hosts sometimes use the expression when introducing big stars.
On Japanese music shows, big stars usually don’t sing first and their fans have to wait long until they appear. TV hosts would say this to excite the viewers.
(OMATASE ITASHIMASHITA. “ARASHI” SAN DESU!)
= They are (finally) here/You must have been waiting for them. It is “ARASHI”!
I sometimes say おまたせしました to my family when I serve dinner. I don’t mean to apologize or thank them for waiting. It just sounds nice. The translation can be similar to yours in that situation.
I hope it helps.