Why does quantum physics seem so weird, even to physicists?

Why can a particle act like a wave?

Particle and wave are familiar words borrowed from larger things. An electron is not a little water wave, and it need not switch costumes between a ball and a ripple. A quantum description predicts both localized detections and patterns across many detections. The Department of Energy describes this combination as wave-particle duality and uses a wave function to represent possible measurement outcomes. DOE: quantum mechanics.

What does the two-slit experiment show?

Send electrons through an apparatus with two narrow openings and record where they arrive. Each electron produces an individual detection. After many arrivals, the distribution can show interference: alternating places with more and fewer detections. In an experiment reported in 2013, researchers controlled the openings and recorded a diffraction pattern building from individual electron events. The pattern was not just a crowd of electrons bouncing off one another. Original electron experiment.

The surprising comparison is between both openings available and the openings tested separately. Quantum predictions do not generally come from simply adding the two separate detection distributions. The calculation first combines probability amplitudes, mathematical quantities that can reinforce or cancel, and then obtains probabilities. Richard Feynman's lectures explain that distinction. It is why the ordinary picture of a ball taking one of two otherwise unchanged routes gives the wrong prediction for this experiment. Feynman: quantum behavior.

Does looking at it cause the change?

A measurement is a physical interaction with apparatus, not necessarily a person watching. Obtaining information about which opening an electron used changes the experimental conditions. A NIST-hosted research paper models how acquiring that path information is associated with losing interference. The amount and kind of information matter; the slogan observation changes everything is too vague to specify an experiment. Nothing here requires a conscious mind to create the result. NIST: measurement and interference.

What remains mysterious?

Predicting the distribution is a different task from supplying a familiar picture of what happens between preparation and detection. Feynman emphasizes the probabilistic prediction even when the experimental setup is specified. We should distinguish the measured results, the calculation that predicts them, and an interpretation of that calculation. A picture is helpful only while it preserves the predictions.

Confidence is high in wave-particle behavior and interference. This answer does not select a final interpretation of quantum reality. A reproducible experimental disagreement with the calculated probabilities could change the theory's tested scope. For two related questions, see whether entanglement sends faster-than-light messages and quantum claims about consciousness.