Ah does indeed mean amp Hours, that makes your battery 384wH, a fairly reasonable size (better than most 24v bikes).
It's unusual for a display to show Ah, but if it does 4Ah would in simplest theory mean you had used 3/4 of battery. In practice you probably have a lot less than 1/4 left for a couple of reasons.
When the battery is full it will be at somewhere around 28 volts;
by the time it is getting to 1/4 it will only be at around 20v.
So using 3 amps at the start is giving you nearly 50% more power than using 3 amps at the end.
Also, as you use power from the battery the voltage drops still more (voltage sag), especially higher power.
If you stepped up to 15 amps the 20v could easily drop to 18v or even 16v.
That's pretty significant, and if the battery gets to too low a voltage it will cut out altogether to protect the battery from damage. That means that a battery with just a small amount of charge left can only deliver it pretty slowly. If you have a significant hill to climb it may cut out just when you need it most, just as you are tired and nearly home at the end of a long ride. (We are lucky from that point of view, we live near the bottom of a hill.)
I believe the basics of what I have said there are correct. Somebody else with more knowledge may be able to give more precise idea of the exact voltages. They vary from battery to battery. As a battery gets old the total Ah/Wh it can hold is much less than when it was new, and the drop at the lower end particularly severe.