Lewis explains in his question that he was filming something with a camera while standing beneath some powerlines but when he watched the footage back, it was humming. How does this work? The answer is something called electromagnetic induction and it was discovered by Michael Faraday (the man who used to be on the £5 note).
The first thing to do is get our heads round a very unusual idea called the electromagnetic field. Imagine a fish sitting inside a lake. In every direction the fish is surrounded by water, but the fish is probably unaware of it. But if something moves through the water, it will create ripples which the fish will be able to feel. In this way, it's possible to send a signal through the water by disturbing it slightly.
Even though it sounds like made up mumbo-jumo, our Universe appears to be filled up with a similar type of fluid. In every direction we are surrounded by an invisible force-field which goes on forever. We call this invisible substance the electromagnetic field. Whether it's made of anything or whether it's fundamentally pure, we have no idea. But it's everywhere and certain things can interact with it, creating ripples and bends in the field. A magnet, for instance, distorts the shape of the field so that certain metals will be twisted and pulled in along with it, following what are called the "field lines".
One of the particles which seems to interact particularly well with the EM field is an electron. As an electron moves around in the Universe, it creates ripples in the electromagnetic field as it goes. They're usually quite faint, but if we get enough electrons all moving in the same way, we can create an enormous electromagnetic ripple which travels through the Universe and can be detected by distant electrons. I know a lot of this sounds like nonsense, but this really is hard Science.
Now let's talk about powerlines. Electricity is usually the movement of electrons through a substance (there are some exceptions but this covers 95% of cases). As electrons stream down a powerline they create an electromagnetic "wake" behind them, the same way a speedboat creates a wake on the surface of an otherwise sleepy lake. This, on its own, doesn't do anything exciting, but if you get the electrons to zip back and forth inside the wire it creates waves in the EM field - waves which travel.
Electricity in a UK powerline is made up from trillions upon trillions of electrons hopping backwards and forwards at 50 Hertz (50 times per second). This means the electrical current in a powerline is generating EM waves 50 times a second.
Inside all electronic equipment there are more electrons zipping around inside its circuitry. For the most part they stay on course, but if a big enough EM wave approaches, the electrons inside the device will start to oscillate as well. This means that anythinig which is recording the world will pick up these 50 Hz EM waves. Having the camera near a powerline is doing exactly that.
The electrons in the powerline are zipping back and forth, this creates EM waves which radiate out from the cables. The waves reach your camera and cause a faint waggling back and forth of the electrons inside, causing a faint buzz or hum. The magic of Electromagnetic induction!