In his famous Lectures on Physics (some of the lectures were repeated in the 1967 Messenger Lectures at Cornell and published as The Character of Physical Law), Feynman famously said that "nobody understands quantum mechanics" and that the two-slit experiment contains "all of the mystery of quantum mechanics."

I will take just this one experiment, which has been designed to contain all of the mystery of quantum mechanics, to put you up against the paradoxes and mysteries and peculiarities of nature one hundred per cent. Any other situation in quantum mechanics, it turns out, can always be explained by saying, 'You remember the case of the experiment with the two holes? It's the same thing'. I am going to tell you about the experiment with the two holes. It does contain the general mystery; I am avoiding nothing; I am baring nature in her most elegant and difficult form.

The Character of Physical Law, chapter 6
Watch the Video

In some of the more accessible material from Lectures on Physics re-published as Six Easy Pieces, Feynman argued that the most important scientific knowledge - from physics to biology - is the simple fact that all things are made of atoms.

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied...
(Six Easy Pieces, p.4)

Everything is made of atoms. That is the key hypothesis. The most important hypothesis in all of biology, for example, is that everything that animals do, atoms do. In other words, there is nothing that living things do that cannot be understood from the point of view that they are made of atoms acting according to the laws of physics. This was not known from the beginning: it took some experimenting and theorizing to suggest this hypothesis, but now it is accepted, and it is the most useful theory for producing new ideas in the field of biology.
(Six Easy Pieces, p.20)

Feynman is quite right that everything is made up of discrete particles. We might rewrite his advice to the future this way:

The universe consists of discrete, discontinuous, and in some sense "digital," particles. There is no "classical" world, only a quantum world. The "classical" world emerges from the quantum world when a large enough number of particles get together. The continuous space (and time) in which we locate the particles is but a mathematical construct that allows us to describe the world.There are no continuous "fields" in which particles of matter (electrons, atoms, etc.) are thought to be singularities. The continuous, causal "forces" like gravity that we postulate are useful fictions. They are only statistical averages over other types of particles (photons, bosons, gravitons) that look continuous when very many such particles are present. At the microscopic level, quantum events are discontinuous and acausal. The analytic integral and differential equations that we assume deterministically govern the motions of material particles are idealizations only accurate for very large bodies.

The idea of a single path for a quantum system (for example, the path of an electron or photon in the two-slit experiment) is replaced with a sum over an infinity of quantum-mechanically possible paths to compute a probability amplitude. It corresponds to the wave picture of spherical waves going in all directions that was critically questioned by Albert Einstein in his 1905 and 1909 papers on the light-quantum hypothesis and wave particle duality.

More importantly, the square of the complex probability amplitude gives us the same probability of finding a particle somewhere as the absolute modulus of Erwin Schrödinger's wave function |ψ|² which is the solution to the Schrödinger equation.

As Feynman wrote in his 1948 article on the new path integral formulation of quantum mechanics...

Non-relativistic quantum mechanics is formulated here in a different way. It is, however, mathematically equivalent to the familiar formulation. In quantum mechanics the probability of an event which can happen in several different ways is the absolute square of a sum of complex contributions, one from each alternative way. The probability that a particle will be found to have a path x(t) lying somewhere within a region of space time is the square of a sum of contributions, one from each path in the region. The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of h) for the path in question. The total contribution from all paths reaching x, t from the past is the wave function ψ(x, t). This is shown to satisfy Schroedinger's equation.

"Reviews of Modern Physics 20.2 (1948) p.367.

Lecture 1 - The Law of Gravitation (video only)

Lecture 2 - The Relation of Mathematics to Physics (video only)

Lecture 3 - The Great Conservation Principles (video only)

Lecture 4 - Symmetry in Physical Law (video only)

Lecture 5 - The Distinction of Past and Future (text and video) (video only)

Lecture 6 - Probability and Uncertainty (text and video) (video only)

Lecture 7 - Seeking New Laws (video only)

In his sixth Messenger lecture, Feynman imagined a scenario like that Arthur Holly Compton used as a model for free will based on quantum uncertainty.

...we could cook up — we'd better not, but we could — a scheme by which we set up a photo cell, and one electron to go through, and if we see it behind hole No. 1 we set off the atomic bomb and start World War III, whereas if we see it behind hole No. 2 we make peace feelers and delay the war a little longer.

Isn't it strange and somewhat diabolical the kind of examples some physicists come up with?

Feynman on the Two-slit Experiment
Probability and Uncertainty - the Quantum Mechanical View of Nature

Feynman on Irreversibility
The Distinction of Past and Future

1965 Nobel Prize Lecture
The Development of the Space-Time View of Quantum Electrodynamics

Finally, Feynman famously claimed that "No One Understands Quantum Mechanics". And he said the two-slit experiment is the one and only mystery in quantum mechanics.

We believe Feynman said "no one understands" because there is no "machinery" that can explain how the abstract and immaterial mathematical wave function can cause the material particles to appear at the locations predicted by the square of the wave function.