[verb ない → なくて] + すみません (apologizing for not doing)
The structure ~てすみません is used to apologize for a specific action you have done (or are doing). It combines the て-form of a verb with すみません ("I'm sorry") to express regret about that action. It translates naturally as "I'm sorry for ~ing."
When apologizing for something that has already happened, you can use the て-form of the past (~てしまって) or simply the て-form, since context usually makes it clear.
When apologizing for not doing something, you use the negative て-form 〜~なくて + すみません.
For more formal situations, すみません can be replaced with すみませんでした (emphasizing a completed past action) or the more formal もうしわけ申し訳ありません / もうしわけ申し訳ございません.
In casual speech, すみません is often replaced with ごめん or ごめんなさい:
Be careful not to confuse apologizing with thanking. Where English speakers might say "sorry for the trouble," Japanese speakers sometimes prefer to thank instead: てつだって手伝ってくれてありがとう ("Thank you for helping me") rather than apologizing for causing the effort.