“Sorry to keep you waiting.”
You're meeting your friend for lunch at a restaurant. You show up 5 minutes late. You say this to apologize to your friend for being late.
Sorry to keep you waiting.
Want Video and Sound? Follow us on YouTube
Sorry to (do something)
This is a way to apologize for a small mistake you made. When you make a larger mistake and want to apologize more strongly, you can say "I'm sorry for ___ing":
I'm sorry for making you wait so long.
But being 5 minutes late to a lunch appointment is not a big mistake, so saying "Sorry to keep you waiting" is a perfect way to apologize.
keep (someone) waiting
To "keep someone waiting" means to make them wait for a while. Use it like this:
They kept me waiting for over an hour. I was furious!
When you use "keep" in this way, it means to make something continue. So to "keep" a person waiting means that you've made them continue to wait. Another example is "keep (someone) guessing" which means that you do a lot of confusing things, so the person has to continue to guess what you're going to do next.