"I'm very sorry" is a very formal way to tell someone that you're sorry. Here's how it sounds in different situations:

In a situation like the one above, it sounds polite. It's polite to use formal English with customers and people you don't know well.

With close friends and family members, it can sound like you're angry if you use overly formal English. So if you tell your close friend:

I'm very sorry.

...your friend may actually think that you're not sorry at all! It's better to say something like:

I'm really sorry.

I'm so sorry!

However, if you're talking to someone close to you AND it's a formal situation (like if you're apologizing in front of a large group of people), you can say "I'm very sorry."