Does anything exist in the intergalactic space?
As others have said, it's almost empty, but not quite, as there are gas particles and so on floating around. As wikipedia states:
Generally free of dust and debris, intergalactic space is very close to a total vacuum. The space between galaxy clusters, called the voids, is probably nearly empty. Some theories put the average density of the Universe as the equivalent of one hydrogen atom per cubic meter. The density of the universe, however, is clearly not uniform; it ranges from relatively high density in galaxies (including very high density in structures within galaxies, such as planets, stars, and black holes) to conditions in vast voids that have much lower density than the universe's average.
And that's only if you consider empty to mean void of matter - there's also electromagnetic waves permeating most (all?) of space. And when you get down to the subatomic level, quantum mechanics ensures that particles are constantly popping into and out of existence as well, even in 'empty' space.
As for how the matter got there, well aside from the normal ways (being shot out of exploding stars and so on), don't forget that before it all started expanding, all of the matter was in the same place anyway, so the particles in intergalactic space haven't necessarily travelled anywhere to get there. They could have simply stayed where they were while particles around them got gravitationally drawn into nearby clumps of matter/galaxies.
I might add some further notes to the actual material things existing in intergalactic space. One might wonder but the notion that there is space is already stating that there is more than nothing.
It implies that there is at least vacuum which is a pretty interesting thing on its own.
Quantum Mechanical harmonic oscillator
Maybe you know that the harmonic oscillator has energy levels
$E_n = \hbar \omega \left( n + \frac{1}{2}\right)$
and an astonishing result is that the lowest energy state is $E_0 = \frac{1}{2}\hbar\omega > 0$.
Quantum electrodynamical oscillator
Coming back to the vacuum, the situation is somewhat comparable. Considering Heisenberg's Principle of Uncertainty in its energy-time form,
$\Delta{t}\cdot\Delta{E} \geq \hbar$
we can see already that a state of a quantum system with definite zero energy for all times cannot exist, even though the expectation value might vanish.
Going more into detail, we see that the operator of the vector potential fullfills the wave equation
$\Delta{A_l} - \frac{1}{c^2}\partial_{tt}A_l = 0$
and a Helmholtz equation if one puts $\partial_{tt}\rightarrow{-\omega^2}$. This equation is usually tackled by separation of variables and after some math we arrive at a Hamilton
$H = \frac{1}{2}\sum_{\lambda}\left({p^2_\lambda+\omega_\lambda^2\lambda{q^2_\lambda}}\right)$
where now $\lambda$ accounts for some mode index. And here comes the magic. This is a description equation for harmonic oscillators! But here we run into a conceptional difficulty. The vacuum energy
$E_{vac} = \frac{1}{2}\sum_\lambda{\hbar\omega_\lambda}$
is infinitely large since there are infinitely many modes of the vacuum. But this is not very physical, so most of the time for calculations you just "leave out" this part.
Implications of a vacuum energy
In the case of different separated domains where you are able to allow a different different number of modes (e.g. via metal plates), this energy will be different for those domains resulting in a force which is the famous Casimir effect.
But vacuum energy has other implications. One hope it that it might some day explain the cosmological constant in terms of a unified field theory.
So, I hope, I could convince you that "empty" might be much more one would expect :)
Sincerely
Robert
At higher redshifts, we actually see huge clouds of neutral Hydrogen (atomic, not molecular) in the form of absorption lines from distant quasars, called the "Lyman alpha forest"
These clouds may eventually form galaxies and stars, but they are currently just gas, an not particularly low density either.