When it comes to "regulating" an output, what the driver circuitry does is regulate the output voltage such that the desired constant current is achieved........if it is a regulated current output type of driver which are the best.
An LED's brightness is determined by the amount of current being passed through it. But before any current will pass through an LED, the voltage on the LED has to be at or slightly above its forward voltage (Vf)........which is dependent on the model of LED........but its around 3 volts. At that point the LED starts to turn on but it doesn't take very much more voltage for the LED to actually even be damaged if the source can provide the current.
It's not a linear response. It's a fairly tight knee from off to fully on. So the best way to control the brightness on an LED is to manage the current through it which lets the driver circuit put out whatever voltage it needs to, to meet this current.
When you see voltage requirements of drivers its the maximum or minimum required input voltage such that the driver circuit works properly. Integrated circuits that operated down at very low voltages tend to be such that they are not tolerant to high voltages. Thus some of these driver circuits are very specific that you do not exceed their maximum voltage input. Being too low just means it won't work properly.
Hope this helps.