First a word of caution: The information in public science videos like this is often like a game of telephone: A researcher tried to explain his/her results to a science communicator in lay terms, and then the communicator tries to further improve on the analogies to make the video more intuitive or appealing. The final product often has a tenuous connection to the original result.
In the present case the statement in the video is based on visualizations of numerical results obtained in numerical simulations of QCD (quantum chromo dynamics). What is observed is that in the QCD vacuum there are large fluctuations of the gluon fields. When a very heavy quark-anti-quark pair is added one observes the formation of a coherent gluon field that connects the quark and anti-quark, a "flux tube" or "color string". Researchers have carefully measured the energy per unit length of this string (called the "string tension"). In particle physics units it is about 1 GeV/fm (giga electron volts per femto meter). Note that constant energy per length means constant force, and we often say to laymen that the force is about a tonne.
There is no analytic understanding of this phenomenon (indeed, a closely related QCD problem, the mass gap, is included among the Clay Millenium problems).
There are some related or analogous systems in which flux tube formation can be understood.
The most famous is a type II superconductor. Electron pairs (carrying electric charge) condense, and the photon acquires a "mass" (strictly, a finite screening length) by the Meissner-Higgs effect. It is then energetically favored for an external magnetic field to be channeled into narrow flux tubes. Inside the flux tube the condensate vanishes and the photon is unscreened. This implies that if magnetic charges existed (they don't), two magnetic monopoles inserted into a type II superconductor would be connected by a flux tube.
A model of what happens in QCD (that can be made precise in certain cousins of QCD not realized in nature) is that the QCD vacuum is a type II superconductor in which the role of electric (color) and magnetic (color) charge is reversed (sometimes called a "dual" superconductor). Composite color magnetic monopoles are condensed, and as a result color electric objects (quarks) are connected by flux tubes.
Finally, in what sense do flux tubes suppress vacuum fluctuations? The video is a little unclear here, but I think what they mean is the following: Ordinary vacuum fluctuations lower the ground state energy of the QCD vacuum (a QCD Casimir effect, or a QCD "cosmological constant"; technically, the QCD trace anomaly). Since a flux tube has positive energy relative to the QCD ground state, I can think of it as having removed (negative energy) vacuum fluctuations.
P.S.: The animations in the video are from this website, and technical papers are here.