Firstly, the image you provided is not of the BFR, but the BFS (Big F***ing Spaceship) or ITS (Interplanetary Transport System) spaceship (Musk is purportedly not happy with either name).
Elon Musk answered a question in his Reddit AMA on r/SpaceX recently that describes the exact purpose of the spheres. Long story short, yes, they are propellant tanks. They are not so much suspended but held in place inside the tanks. Enough of me talking though.
ITS Spaceship design question II.:
The ITS Spaceship has two mystical spherical tanks, marked green in this slightly edited image. The whole tank design looks very exciting, and there's rampant speculation on this sub about the purpose of those spherical tanks:
are they for landing fuel?
... or are they storing 'hot' gaseous propellants as part of the autogenous propellant pressurization system?
... or are they used for on-orbit propellant densification to store vapor before it's liquefied again?
All of the above perhaps?
Those are the header tanks that contain the landing propellant. They are separate in order to have greater insulation and minimize boil-off, avoid sloshing on entry and not have to press up the whole main tank.
As an addendum, SpaceX's Interplanetary Transport System does not use Helium, instead opting for autogenous pressurization whereby you use the propellant to pressurize itself (gaseous Methane and gaseous Oxygen, respectively).