When the superfluid helium leaks out of the container, that's not tunnelling. A few drops of superfluid helium contain many many atoms and for all of them to tunnel through would be extremely improbably as to pretty much not happen. So what makes the helium flow out?
If you notice carefully, the container is not a normal glass mug, but it's bottom is made of some other material (I think it's some kind of ceramic with tiny pores). As the scientist in the video says, the only reason "normal" liquid helium (above the critical temperature) did not flow out through those pores, was that viscous drag kept it from dripping out. After the phase transition to a superfluid, the *super*fluid doesn't have any viscosity and no longer resists flow. So it flows out of the container.
As far as the wall-climbing goes: It should somehow be that (loosely) it is energetically/probabilistically preferable for the fluid to climb up the walls and then flow out rather than stay in. I don't really understand why exactly it climbs the walls of a container and a cursory search seems to indicate that though there are explanations based on viscosity for this too, there isn't any clear consensus which is perfectly validated by experiment.
I think this video (with annotations) is more informative: https://www.youtube.com/watch?v=2Z6UJbwxBZI