Your interpretation is correct if acceleration is constant and the motion is in a straight line. The object will change it's velocity by that much every second.

Quick example. If you drop an object, it's acceleration will be about $9.8~\text{(m/s)/s}$. This means after one second it's traveling at $9.8~\text{m/s}$, after two seconds it's traveling $19.6~\text{m/s}$, and so on.

As a side note, most often people "do math" on the units so that (m/s)/s is written m/s$^2$. This hides the interpretation of acceleration, though. Your way of writing it is more clear, and just as correct.

(The interpretation gets a little trickier if acceleration is not constant or not in a straight line. In the latter case, one could have a constant speed but a changing velocity due to direction change. But the interpretation still has to do with a change in velocity per unit of time.)

Your interpretation is correct if acceleration is constant, and the motion is in a straight line. The object will change its velocity by that much every second.

A quick example: If you drop an object, its acceleration will be about $9.8~\text{(m/s)/s}$. This means after one second it's traveling at $9.8~\text{m/s}$, after two seconds it's traveling at $19.6~\text{m/s}$, and so on.

As a side note, most often people "do math" on the units so that (m/s)/s is written m/s$^2$. This hides the interpretation of acceleration, though. Your way of writing it is more clear and is just as correct.

(The interpretation gets a little trickier if acceleration is not constant or not in a straight line. In the latter case, one could have a constant speed but a changing velocity due to direction change. But the interpretation still has to do with a change in velocity per unit of time.)