The grammar pattern てすみません is used to apologize for (A). A common translation is 'I'm sorry for (A)'. This structure utilizes the causal meaning of the て-form, which can indicate 'because of' or 'as a result of'. Let's examine some examples.

• 大事(だいじ)な書類(しょるい)を忘(わす)れてすみません。
  I'm sorry for forgetting the important documents.
• あなたの携帯(けいたい)を壊(こわ)してすみません。
  I'm sorry for breaking your cell phone.

Sentences using the (A) て (B) structure often imply that the outcome (B) was an unintended consequence of the action (A). This grammar point originates from the verb 済(す)む, which means 'to be concluded' or 'to be settled'. Its negative form, すみません, conveys the feeling of something being 'unsettled'. As a result, てすみません literally suggests 'because of (A), this matter is unresolved', which is much like the English concept of feeling indebted or that an obligation is unfulfilled.

• 部屋(へや)を汚(よご)してすみません。
  I am sorry for making the room messy. (It unfortunately resulted in the room being left in an unclean state)

While it is typically translated as 'I'm sorry', it may be more useful to think of this grammar pattern as a way of emphasizing the speaker's personal regret or sense of unresolved obligation regarding (A), and conveying this feeling to the person they are addressing.