Good question! It means that the probably of finishing in the next increment of time is always the same. Just got up to the teller? There's, say, a 20% chance that you're done within the next minute. Been at the teller's desk for 30 minutes already? Still a 20% chance that you're going to be wrapped up in the next minute.

This might be saying the same thing in a slightly different way. But does it also mean that if we plot "service times" on the X axis and frequency on the Y, then in the first quadrant we get something high on the left, getting lower as it goes right, and asymptotically approaching zero as it goes to infinity on the right?

It means that the service time is modeled as an exponential random variable. See https://en.m.wikipedia.org/wiki/Exponential_distribution .

Think about radioactive half-life. Half the atoms will decay in X minutes. Half the remainder in 2X minutes. Half the remainder in 3X etc.

Likewise you'll serve half your customers within X minutes. Half the remainder within 2X, half the remainder within 3X etc.