You could have a dimmer that goes from one level of lightness, up to another, with 8 steps in between. And you could have another dimmer that goes between exactly the same two levels of lightness. The same range. but with 1,000,000 steps in between. The range of the two dimmer switches is the same. They both go from the same minimum to the same maximum. Just the number of steps in between is different. The number of bits represents the number of steps.
Why is 1,000,000 steps the required number? 1,000,000 is just the number of integral factors in the range in base 10. But why is an integral factor more important than a fractional one? They could both be perceptible to a human. The range is the same in both cases, and the given explanation was given that the range was so high.
The thing is that you can't set the minimum value arbitrarily, because the scale is anchored to zero. So "1" is always one step above zero, and the step size is a function of bit depth.
So going with our example here, a dimmer with 1000000 steps would not have the same range as dimmer with 8 steps, because the the minimum brightness of one is 1/1000000th of max, and 1/8th of the max for the other (assuming linear scale for simplicity's sake, the principle applies to nonlinear scales too)
Imagine you have a lightbulb (producing let's say max 1000 units of light/brightness [which are two subtly different things]) connected to a dimmer that has for example 8 discrete levels. What would you say the dynamic range of this setup would be? Obviously measuring the contrast ratio from when the dimmer is in the zero setting (=light off) to max setting would result infinite contrast ratio/dynamic range. That is not very useful though, so the next best thing is to measure brightness difference from the lowest on setting ("1") to max, which gives you actually a reasonable number, in this scenario 125:1000 or 1:8. I hope this example was illuminating.
So I guess the idea is that bit depth fundamentally determines what is the minimum brightness representable without going to zero (where be dragons), and as such determines the max usable dynamic range.