This pattern is used to apologize for an action or situation that you caused or are responsible for. It is a polite, everyday expression. The te-form shows the reason or cause, and sumimasen conveys the apology. It often carries a sense of gratitude mixed with apology when you inconvenience someone, but in this structure it is primarily apologetic.

You can use it with verbs to apologize for your own actions, like arriving late or making a mistake. With adjectives, it apologizes for a state, like being slow. With nouns or na-adjectives, use で instead of て, as in 病気で (being sick).

This is a spoken, polite form. In very formal situations, you might hear て申し訳ございません, but てすみません is natural in daily conversation, including with coworkers and acquaintances.