A sit-up is a kind of exercise. The joke is that he "sits up" in bed to hit a button on his alarm clock so he can go back to sleep. He's saying that he would do more of those sit-ups if he could, but the alarm clock only lets him hit the snooze button (the ten-minute delay button) five times.
Note that "there's" isn't grammatical. It's a common mistake. It should be "there are."