Because we have this formula in the forward direction:

z{[l]} = W^{[l]} \cdot a^{[l-1]} + b^{[l]}

What happens when you differentiate? Note also that we are taking derivatives of the cost, so that is “in the numerator” if we can use that slightly “off kilter” terminology (a derivative is not really a fraction, but I hope you see what I mean there). Remember what Prof Ng means by his “d” notation for gradients:

db^{[l]} = \displaystyle \frac {\partial J}{\partial b^{[l]}}

Note that Prof Ng specifically designed these courses not to require knowledge of calculus, so he doesn’t explain this type of derivation. If you have the math background, here’s a thread with links to lots of information on the derivation of back propagation.