This phrase is used to apologize for something big or small that you have done, missed, etc. This phrase can also be expanded with the conjunction “for” to say what you are sorry for. E.g. :
- I’m sorry for being so late.
- I’m sorry for the mess. I wasn’t expecting anyone today.
If you add “really” to this phrase, you are saying that you are very sorry about something.
- I’m really sorry I didn’t invite you to the party.