We have a terrible "everyday understanding of acceleration":
I think your real problem is that we don't have a very good everyday understanding of acceleration. We spend most of our time going about the same speed. The one place we do commonly think about acceleration is cars. High-performance cars will often brag about their acceleration like "goes from 0 to 60 mph in 6.2 seconds". They mix time units, using both hours and seconds, "miles per hour" is distance / time, and "in 6.2 seconds" is 1 / time. We can put this in Google to translate it to 5.58 m/s/s, but that's a much harder-to-interpret number (especially for an American!).
But in "0 to 60 mph in 6.2 seconds", hopefully, the units do make sense. From a stop, it takes 6.2 seconds to get up to freeway speed. Stopping a car is another place where we think a lot about changes in velocity, but even there we don't tend to focus on (de)acceleration, rather we focus on the distance covered while stopping---which would require a couple integrals to calculate based on acceleration data!
A little extra info: the rate of change of acceleration is jerk, which always makes me think of being on a roller coaster. Often, at the end of a roller coaster, as you pull up to the loading platform the cars are lightly braked, so your decelerating just a little bit. Then they put on a hard brake and stop you, which very quickly gives you a big negative acceleration, then no acceleration (big jerk). And usually the jerk bumps your head against the headrest.
Explanation with units:
Let's define
Rate of Change: amount of change divided by length of time for change.
Whatever units you use to measure a quantity, those are the units used for measuring differences in those quantities.
For velocity, which is the rate of change of position, we use meters to measure position. If you go 20 meters in 5 seconds, then your (average) velocity is 20 m / 5 s = 4 m/s.
Pretty much whenever time is in the bottom of a fraction, you've got a rate of change for whatever else there is. This way, m/s is a rate of change for meters (position).
Velocity, as we said above, is measured in m/s. If you're going 4 m/s, and then 10 seconds later you're going 9 m/s, your velocity has changed so clearly you've accelerated! Subtraction tells us the change in velocity, 9 m/s - 4 m/s = 5 m/s, but to get the rate of change we need to divide by the time it took for the change to happen: 10 seconds. 5 m/s / 10 s = 0.5 m/s/s.