Use this phrase when you have to give someone information that they won't like. For example, a waiter can say this to a customer:

I'm very sorry, sir, but I'm afraid we're all out of the salmon.

You can use this phrase to say "no" to a request:

A: Excuse me, can you spare some change?

B: I'm afraid not.

"I'm afraid..." is a little bit formal.

Notice that this phrase doesn't actually mean that you're frightened; it's just a polite way to show that what you're going to say will be disappointing to the listener.