The exponential distribution is the distribution of the waiting time until an "arrival" when the conditional probability that the arrival happens in the next minute, or the next microsecond, or the day month, given the amount of time you've already waited, does not depend on the amount of time you've already waited. It's just as likely that the next phone call will arrive at the company's switchboard in the next ten seconds, after you've waited three minutes, than it is that the next phone call will arrive at the company's switchboard in the next ten seconds after you've waited two seconds.
If, on average, there is one arrival every three minutes, then the rate is $1/(3\text{ minutes})$ or $1/3$ per minute. It's just the average frequency of arrivals. If there are on average ten arrivals per minute, then the rate is $10$ per minute.