A derivative is the rate at which one thing changes as another changes. The chain rule says that if A affects B and B affects C, you can work out how A affects C by multiplying the effects together. That single rule, applied across many layers, is the whole mechanism by which a deep network learns.
Technical view
Backpropagation is the chain rule applied in reverse mode, computing all partial derivatives of a scalar output in one backward pass at roughly the cost of the forward pass. Reverse mode is efficient precisely when outputs are few and inputs are many, which is the shape of a loss function over parameters. Automatic differentiation frameworks implement this by building a computation graph during the forward pass.
Related
Part of AI Terminology Index, section: Maths and Statistics.