What phenomena does "Coherence Bandwidth" explain?
The coherence bandwidth measures how much a channel is statisticallt flat given a fixed size window through which we watch it.

Imagine a transmitter and a receiver, the transmitted signal being \$x(t)\rightleftharpoons X(f)\$. The receiver ideally gets the same \$x(t)\$ at the antenna, but unfortunately there's the mighty channel \$c(t)\rightleftharpoons C(f)\$. The receiver will then get \$Y(f) =X(f)\ast C(f)\$.

That \$C(f)\$ is ideally 1, i.e. it's perfectly flat for each frequency (please note that \$C(f)\$ is a complex function, i.e. \$C(f) : \mathbb{R} \rightarrow \mathbb{C}\$). That's not what happens in real life. If we measure just \$|C(f)|\$ completaly disregarding the phase, that is fundamental in almost all modulation techniques, we find out that it's everything but flat.

Now just close the math-ish door and let all the engineering folks in. There's already one shouting "Hey, if you look close enough that \$C(f)\$ is flat indeed.". Well, he's right and that's where the coherence bandwidth kicks in: you have the channel transfer function, you look at it close enough, and that becomes flat. The coherence bandwidth tells you how much is close enough: if that's 1kHz, well you've got to magnify your \$x\$ axis to see only a 1kHz portion at a time. If that's 1MHz... Well, you guessed it.

So why is it a statistical parameter? Well, you can't certainly measure all the channels you want to transmit on. Some guys at IEEE one day decided "ok, if you are in a city with tall buildings you should expect a \$B_C\$ of this much, if you are on a flat desert \$B_C\$ would be that much", and so on, and various models were born.

When does the delay spread kick in? Well I see you quite grasp what it is, and as you say \$B_C\$ and \$D\$ are close friend. As you know \$D\$ measures how much delay we should expect between the direct (the most direct) path and the others. That number tells us how long before our signal gets compromised by itself. Well it appears that a nice rule of thumb (read: there's little to no physical meaning associated to the following formula) is \$B_C=\frac{1}{D}\$. And we love to know the \$B_C\$ of a channel because it tells us how much bandwidth our signal can use without using advanced techniques such as an equalizer.

Added after OP request
What is delay spread? Here is what I have in my notes:
$$\Delta\overline{\tau}=\sum_{l=1}^Np_l\tau_l$$
Where:


N is the number of paths
\$\tau_l\$ is the delay associated with the l-th path
\$p_l\$ is the normalised power of the l-th path so that \$\sum p_l=1\$


The delay spread hence express something like the delay after what I expect most of the power is arrived at the receiver. Consider the situation with:
$$N=3\\ \mathbf{p}=[0.7, 0.2, 0.1] \\ \mathbf{\tau} = [1,2,5]ms$$
You get \$\Delta\overline{\tau}=1.6ms\$

While if:
$$N=4\\ \mathbf{p}=[0.3, 0.4, 0.2, 0.1] \\ \mathbf{\tau} = [1,2,4,7]ms$$
You get \$\Delta\overline{\tau}=2.6ms\$

As you can see the delay spread measures when most of the power will arrive, and it's quite useful to appropriately tune the receiving equalizer to get the most out of your signal.