Space is a part of the wavefunction, as the article explains clearly. The wave function describes where the particles can be in physical space. And, the wave function has the same shape as the wave equations for traditional mechanical waves, like a sound wave or a sea wave.
However, if a classical three-dimensional wave equation describes how matter osciallates in three-dimensional physical space, a quantum wavefunction doesn't do that. Quantum particles don't oscillate in physical space like that. A three-dimensional wavefunction might describe three particles' positions along a one-dimensional line, and it's oscillations are oscillations of probability, not position. The particles don't move, say, up and down. Their probability to be here or there on that 1-d line waxes and wanes.
This is what the article is trying to explain: the basic mathematics of quantum mechanics, the definition of the wavefunction. The value of a wavefunction for the position of three particles is not a position in space at a moment in time. It is a (complex) probability for the position of every particle at that moment.
This only seems confusing when looking at wavefunctions that describe positions. But wavefunctions often have many more observables, such as spin or polarization. A wavefunctions for two electrons moving around on a plane will not be a two-dimensional wave. It will be a wave in a six-dimensional space, whose axis may be "particle 1 has spin up/down, particle 2 has spin up/down, particle 1 position along x axis, particle 2 position along x axis, particle 1 position along y axis, particle two position along y axis".
I'm honestly confused; it's fine to say the wave function lives in some high dimensional phase space and that it's not actually describing some vibration of spacetime. But I don't recall ever imagining the wave function being a vibration of spacetime, is that really something people think?
If I were to express some sort of wave-function-in-spacetime theory, I'd invoke lots of classical fields filling space and have those wiggle.
In any case, the whole bit about the proper two-particle wave function living in a higher dimensional space is somewhat spoiled by the fact that you can factorise it into normal 3-space pieces (so long as you don't have your particles interacting), it doesn't seem such an alien space to me.
Before the wavefunction, we used to explain the double slit experiment (the version without detectors at a slit) as light being an EM wave in physical space, essentially equivalent to a sound wave propagating through the EM field, which breaks on the wall and essentially transforms into two separate waves, each originating from one slit, which are then in phase and so they constructively interfere, forming the final pattern on the screen.
Lots of people think that this is the same picture that the wavefunction gives, but this is wrong. In the QM picture, the emitter emits one photon, which is a quantum of energy described by a four-dimensional wavefunction which assigns some probability of a detection event at the slits, at the screen, etc. In this picture, there is no physical EM wave, any interaction with the light will happen at a single localized point in space. Of course, if you add more particles, especially those carrying charges, the picture changes, and you'll see probabilities that roughly correspond to a picture of an oscillating EM field. But the wavefunction, which is the "bedrock" physical theory, is separate from those waves in the EM field, which are just an approximate picture of the probabilities dictated by the wavefunciton.
However, if a classical three-dimensional wave equation describes how matter osciallates in three-dimensional physical space, a quantum wavefunction doesn't do that. Quantum particles don't oscillate in physical space like that. A three-dimensional wavefunction might describe three particles' positions along a one-dimensional line, and it's oscillations are oscillations of probability, not position. The particles don't move, say, up and down. Their probability to be here or there on that 1-d line waxes and wanes.
This is what the article is trying to explain: the basic mathematics of quantum mechanics, the definition of the wavefunction. The value of a wavefunction for the position of three particles is not a position in space at a moment in time. It is a (complex) probability for the position of every particle at that moment.
This only seems confusing when looking at wavefunctions that describe positions. But wavefunctions often have many more observables, such as spin or polarization. A wavefunctions for two electrons moving around on a plane will not be a two-dimensional wave. It will be a wave in a six-dimensional space, whose axis may be "particle 1 has spin up/down, particle 2 has spin up/down, particle 1 position along x axis, particle 2 position along x axis, particle 1 position along y axis, particle two position along y axis".