they said that the effect of Heisenberg's uncertainty principle shortly after the Big-Bang made matter (normal & dark) expand in a non-homogeneous way.
Quantum fluctuation of the early universe are introduced in the Big Bang model because of the great homogeneity observed in the cosmic microwave background, of order $10^{-5}$ , no matter at what part of the observable universe one maps the CMB. It was necessary because at the time of decoupling of light/photons from matter, there were parts of the universe which due to special relativity could not exchange energy with other parts and come into a thermodynamic equilibrium that could explain the perfect black body radiation shape, no matter what part one looked at.
The statement you are quoting is the one hand waved to explain the tiny $10^{-5}$ inhomegenuities that are observed, emphasized in the plots for clarity :

Since the model is quantum mechanical, quantum fluctuations will exist by construction of the theory. The Heisenberg Uncertainty is a rule of thumb for the way the quantum mechanical operators commute or not with each other, and in the CMB map are reflected in the detection of more (or less) radiation over the background, (again , at a level differing from uniform by orders of $10^{-5}$). These bumps are the seeds for the clusters of galaxies and galaxies as the universe expanded after the photon decoupling.
What is then the connection between us not being able to know both quantities at the same time and the early universe expanding in a non-homogeneous way?
The position and momentum uncertainty is one of the possible fluctuations, as there are a number of non commuting operators in the quantum mechanical models, to introduce uncertainty in the behavior of the primordial energy density . It is not necessary that a human observer observes , it is enough that interactions are happening , these are the "observers" , and a lot of interactions of the inflaton are happening in the Big Bang Model.