Information is a bit weird as a way to put it, but it may be correct. Still, in case this is very intentional wording, the information about the electrical circuit being lit up will (that is, any photon emitted at the moment and location that the voltage appeared, such as hitting a switch to close a circuit) still travels at
c in all directions. It's only the (main) electric current that travels at 60-80% of
c and along the wire.
Also, the phenomenon being discussed in the Veritasium video is slightly different, and that video sort of mixes up two things to make things more mysterious.
The simple explanation for what is going on in the video is that whenever you have a voltage change in a circuit, such as closing it with a switch, you get a tiny bit of radio waves being emitted radially outwards from that point out *.
In the Veritasium experiment, with a very precise measurement tool, they could detect the tiny radio pulse emitted by the switch at the moment it was closed, far before the main current reached the same point moving along the long wires.
The other thing he was mixing this up with is the Poynting vector, which is a mathematical object that represents the direction along which power (as in, worl, the thing you measure in Watts, not the actual current which you'd measure in Amperes) propagates in an electrical circuit. It's not clear at all that the Poyinting vector is a meaningful aspect of reality and not just some mathematical tool. It does happen to coincide with the direction that the radio waves propagate as described earlier, so some physicists do like to interpret it as a deeper, physically real, thing that is related, but you can just as easily ignore it.
Note that when I say that it's "not real", this is similar to how I would describe the Lagrangian and Hamiltonian of a classical mechanics system. It's an extremely useful mathematical tool and it predicts the behavior of the system perfectly well, and is mathematically equivalent to Newton's laws of motion. But while the laws of motion describe real direct aspects of reality, such as objects having mass and momentum, the Lagrangian doesn't really represent an actual thing in physical reality.
The above comes with a gigantic caveat: in QM, the Lagrangian and Hamiltonian correspond to the wave function, which is, to the best of our current understanding, the real underlying thing behind reality, with particles and waves and things localized being the artifical objects we introduce merely for convenience.
* This is normally completely negligible for DC current, but for AC current, where the current switches direction all the time, a relatively high amount of radio waves get generated radically put from the wires along their whole direction. This can lead to problems such as AC power lines causing significant radio interference. This is also normally described as "parasitic", since the power those waves carry is lost from the transmission line, which is designed to carry as much as possible of the power generated at the source towards the destination.