てすみません is a grammar construction that expresses the speaker's regret about (A). It is often translated directly to ‘I’m sorry for (A)’. As discussed in one of our previous lessons on て, we examined the ‘since’, or ‘due to’ meaning that it can have. Let’s look at a few examples.

• 待（ま）ち合（あ）わせの時間（じかん）に遅（おく）れてすみません。
  I am sorry for coming later than the arranged meeting time.
• 息子（むすこ）さんの髪（かみ）をピンクに染（そ）めてすみません。
  I am sorry for dying your son's hair pink.

Phrases that use this particular (A) て (B) nuance of the て-form primarily express something that is beyond the control of the speaker in the (B) statement. Coming from the verb 済（す）む (to finish), すみません (the negated form) means ‘to be unfinished’. In this way, てすみません literally means that the speaker feels that something is ‘unfinished due to (A)’. This is very similar to phrases like ‘having unfinished business’ in English.

• ご飯（はん）を残（のこ）してすみません。
  I am sorry that I didn't eat all the food. (It unfortunately resulted in that I did not finish eating all the food)

Although the translation is ‘I’m sorry’, perhaps it would be easier to remember this grammar structure as highlighting someone's regret/unfinished feelings about (A), and simply showing that to the person that is being apologized to.