Learn Farsi: خیلی متأسفم.

In Farsi, "I am" is often just a suffix attached to the end of a word. The word for sorry is متأسف (mota'assef). The ending ـم (-am) means "I am." So متأسفم literally means "sorry I am." You don't need to add the separate word for "I" (من) at the beginning.

What is the little symbol on top of the letter in the middle of متأسفم?

That symbol is called a hamza (أ), which appears in words borrowed from Arabic. It represents a tiny catch in your throat (like the pause in the middle of "uh-oh"), but in everyday spoken Farsi, it is usually just pronounced as a smooth "a" sound. You can read the word as "mota'assefam".

I already learned that ببخشید (bebakhshid) means sorry. When should I use متأسفم instead?

ببخشید is used more like "excuse me" or a light "sorry" when you bump into someone, want to get their attention, or ask them to repeat something. متأسفم means "I am regretful." You use it for a more serious apology or to express sympathy, like saying "I'm so sorry to hear that" when someone shares sad news.

Why is خیلی (kheili) at the beginning of the sentence?

خیلی means "very" or "a lot." Just like in English, it comes right before the adjective it describes. Because the "I am" part is attached to the end of "sorry" here, the sentence order is literally "very sorry-am."