Given a few plausible assumptions about the universe its spacetime geometry is described by a solution to the Einstein equations called the FLRW metric. If we know the densities of various types of matter/energy present, e.g. photons/matter/dark energy/anything else, then we can calculate how the expansion of the universe varies with time.
Generally speaking the photon/relativistic matter density is only important for times very soon after the Big Bang. So if there is nothing screwy like dark matter around then the evolution is dominated by the density of matter (that's visible and dark matter), and the key parameter is the ratio of the density to the critical density. We call this ratio $\Omega$, and the geometry is related to $\Omega$ as follows:
$\Omega \lt 1$ - closed universe
$\Omega = 1$ - flat universe
$\Omega \gt 1$ - open universe
Our universe appears to have $\Omega$ very close to $1$ so it appears to be flat. The universe discussed in the video you link has $\Omega \lt 1$ so it is a closed universe, and a closed universe has the spatial topology of a 3-sphere. This does indeed mean that if you draw a straight line in space and continue it for long enough the line will eventually meet itself again.
So far so good, but the main point of your question is asking what is the fate of a closed universe, but that turns out not to have a simple answer. If all the closed universe contains is matter and photons then the closed universe must recollapse. That is, it will expand from the Big Bang, reach a maximum scale factor then recollapse again into a Big Crunch. But if the universe contains dark energy this can change its fate. If the ratio of dark energy to matter is high enough then even a closed universe can expand forever.