In a controlled gate, you might expect the target qubit to be the one that changes. But when the target is prepared in a special state (an eigenstate of the gate being applied), something surprising happens: the target stays the same, and instead the control qubit picks up a phase. The control qubit's measurement probabilities do not change, but the phase it acquired can be revealed by a later interference step. This is called phase kickback, and it is one of the standard mechanisms behind quantum algorithms.

You stamp a document on one side, but the pressure leaves a mark on the sheet underneath. Phase kickback is similar: the operation is applied to the target, but the effect shows up on the control. The analogy is only about side effects -- the real mechanism is a relative phase in the joint quantum state.

If the target qubit is in an eigenstate of the controlled gate's action, the gate acts as a phase factor on the control branch. For example, if the target is in $|-⟩$ (an eigenstate of $X$ with eigenvalue $-1$) and the control is in superposition, then the controlled-X (CNOT) gate multiplies the $|1⟩$ branch of the control by $-1$. The target is unchanged. The control now carries a phase that a later Hadamard can convert into a deterministic measurement outcome. This is the mechanism behind Deutsch's algorithm and phase estimation.