Molecular clouds are the birthing grounds from which stars form, which occurs when said clouds undergo a collapse. They contain a significant fraction of molecular hydrogen ($H_2$), and are generally good places for forming other stable molecules. Molecules and metallicity generally make star formation easier by making it easier to get the non-molecular hydrogen and helium to lose enough energy to collapse into a common core by acting as (kinetic) energy absorbers. Imagine tossing a ping pong ball at a tennis ball: the ping pong ball bounces off at a slightly lower velocity, and the tennis ball barely moves at all from an even smaller change in its velocity. And if the cloud is already "cold", then that's even better: there's less energy your particles have to shed. The presence of molecular hydrogen seems to be particularly important in the collapse of low metallicity clouds, such as would be the case in the early universe. So cold molecular clouds, like infrared dark clouds, should be a great nursery for high mass stars, as you don't have to shed that much energy (relatively speaking) to get things collapsing into a core, and it's got molecules to help it shed what it still needs to.
Exactly how the collapse is predicted to happen depends on the model, and the models depend on the magnetic field of the collapsing cloud. There are two major models.
Quoting liberally from Magnetic Fields
in Molecular Clouds by Richard M. Crutcher (c. 2012) (all quotes will be from this article, until otherwise stated):
In the strong magnetic field theory, clouds are formed with subcritical masses, $M < M_\Phi = \Phi/(2\pi \sqrt{G})$ (Nakano & Nakamura 1978), where $M_\Phi$ is the critical mass, $\Phi$ is the magnetic flux, and $G$ is the gravitational constant. Hence, the magnetic pressure is sufficiently strong to counteract gravity and prevent gravitational collapse. Because the magnetic field is frozen only into the ionized gas and dust, neutral gas and dust contract gravitationally through the field and the ions, increasing
mass in the cloud cores. The magnetic field strength also increases, but more slowly than does mass.
This process is known as (gravity-driven) ambipolar diffusion. [Ambipolar diffusion, or breaking of flux freezing, may also be driven on small scales by turbulence (e.g., Zweibel 1988.)] When the core mass reaches and exceeds $M_\Phi$, the core becomes supercritical ($M > M_\Phi$), collapses, and forms stars. During the collapse, the magnetic field is dragged inward but cannot become strong enough to halt the collapse.
The weak-field theory of star formation says molecular clouds are intermittent phenomena, with short (∼$10^6$ years) lifetimes. Magnetic fields are sufficiently weak that the low-density ISM is supercritical ($M > M_\Phi$). Clouds form at the intersection of turbulent supersonic flows. Generally, clouds do not become gravitationally bound, and they dissipate; those that are self-gravitating form stars in essentially a free-fall time (Elmegreen 2000). Supersonic turbulence will dissipate on roughly the free-fall timescale as collapse of gravitationally bound clouds proceeds (MacLow et al.
1998). Although magnetic pressure cannot stop the collapse, it can dominate turbulent pressure during the late stages of core collapse. An extreme version of weak-field models has the initial field so weak that the medium is super-Alfvenic as well as supercritical (Padoan et al. 2004).
So, the short of the matter is that molecular clouds are important because they are connected to the formation of (high mass) stars. And the magnetic fields of those clouds is important because the exact dynamics of that collapse are dependent upon them. And all of that is important because it gives us a better idea of how high mass stars form, which is expected to have happened much more frequently in the early universe.
It hasn't, however, yet mentioned why the direction of that field may be interesting (it only mentions the flux).
Interstellar dust produces thermal emission and extinction of light from background stars. Linear polarization of this radiation provides a probe of the magnetic field morphology in the ISM, including molecular clouds.
So they're curious about the structure of the interstellar medium (ISM) of the galaxy. The radiation of the dust clouds, including the molecular clouds, provides a way to probe the magnetic structure of the ISM. The paper then recalls how this polarization can arise via radiative torques creating preferential alignments in dust grains, and what this then means.
[T]he grain extinction cross-section is greatest perpendicular to $\mathbf{B}$, so the maximum of the polarized light from background stars is parallel to $B_{POS}$, the magnetic field direction in the plane of the sky. Conversely, maximum polarized emission is perpendicular to $B_{POS}$. The predicted degree of polarization depends very weakly on magnetic field strength, so dust polarization does not directly give the magnitude of the magnetic field.
This describes how they can measure the direction of the magnetic field of a molecular cloud, a topic from your clip. It notes that it's a poor measure of the actual field strength, and we need the flux $\Phi$ to check how a given star formation model matches up with observations.
The paper continues on and discusses some measurements of the direction of the magnetic field of molecular clouds. The results it cites indicated that the field has been observed to be either parallel or perpendicular, and that the direction can change between different parts of the same cloud. So I'm not sure why your clip mentions a perpendicular measurement as surprising; perhaps it's the "more-or-less" that is key, as if there's enough data to suggest a preferred (perpendicular) alignment then that would be interesting and probably unexpected.
The paper also discusses how it is nevertheless possible to obtain some estimates of $\Phi$, as well as the difficulties in doing so accurately. This lets us tie everything together: the polarization of the radiation from the molecular cloud allows us to measure the direction of its magnetic field relatively easily; and with that and some various techniques we can then measure the actual flux $\Phi$ of the magnetic field; which we can then use to determine if the cloud is sub-critical ($M<M_\Phi$) or not. And when we know that, we can now more readily probe how accurately star formation models, such as those mentioned above, reflect the actual reality.
