1. Roger had been trying hard to arrange a business deal between two groups. He was still unsuccessful after weeks of trying, so he decided to throw in the towel and spend his time on something else.
2. I’ll never be able to pass my mathematics course, no matter how much time I spend studying. I’m going to throw in the towel and withdraw from the course.

The expression comes from the game of boxing. When a boxer has had enough and is ready to give up the fight, his or her manager throws the towel used to clean his or her face during the fight into the ring as a signal that the fight is over.