Simple intuition
Two possibilities can reinforce each other or cancel each other. Quantum algorithms work by steering the state so the useful outcomes get reinforced.
Precise explanation
A Z gate changes only the relative sign between basis amplitudes. That sign does not change the immediate measurement probabilities in the computational basis, but a later Hadamard mixes the amplitudes and turns the sign difference into a measurable change. That is interference.
Example or analogy
Analogy: water waves can meet crest-to-crest or crest-to-trough. The analogy is helpful for the add-or-cancel idea, but the quantum state is still a complex amplitude, not a literal fluid wave.
Common misconception
Interference is not about particles bumping into each other. Even one isolated quantum system can interfere with itself because its amplitudes add as complex numbers.