"On the clock" is a way of saying "I'm at work" or "I'm working". It is a slang or colloquial expression. It can emphasize that you are busy and do not have time to pause, or it can simply mean you're working. It is "on the clock", because you have "clocked in" to work. "Clocked in" means you arrived at work, looked at the clock, and made a record of what time you arrived and began working. You must continue to work until your working hours are completed, hence you are "on the clock" until your working hours have passed.