In an M/M/1 Queue, what does exponential distribution of service time mean? I was reading about the M/M/1 Queue and that we assume new customers arrive according to a Poisson distribution, and each customer takes an amount of time to service that is drawn from an exponential distribution?

I understand what it means to have an arrival rate based on Poisson distribution - if an average of 10 come per hour, you'll have 9 a few times, 6 very infrequently, 11 pretty often etc - but what does it mean that service is exponential?

Does this mean that as you serve more customers, the average time it takes to service them rises exponentially? Wouldn't the system crash soon after a few customers?
In an exponential distribution, the probability is DECREASING exponentially -- hence little risk that service time will grow. On the contrary, the mean service time is fixed, at the inverse of the rate.
This distribution is actually very rare in practice, but is chosen because it is Markovian:
the probability to terminate servicing a client does not depend on the past, and in particular on how much time you already spent servicing that client. That yields very nice mathematical properties.