What is the purpose of the 3 levers? Why not a single lever, or even just a single button?
The set designer thought three levers looked cooler.
They actually control the three seashells.
One controlled the transporter sound, one controlled sparklyness, and the last was the Good/Evil control.
They went back to analog controls after discovering the hard way that pixelization is not something you want to happen to living beings.
The GUI Interface was too confusing, and usually resulted in the dreaded BSOD. Back to analog.
X, Y and Z coordinates.
Sure it’s tricky, that’s why you need an engineer to operate the things.
They do nothing at all, except to give the dude running them some feeling of control at the time, especially when something is going wrong.
That’s the beauty of it. They don’t do anything!
When deconstructing a physical organism and reconstituting it from an atomic level, there’s nothing more reliable than executing such an exacting process manually, rather than with an untrustworthy computer.
I always figured they were power controls. I mean the thing eats a lot of energy.
Just be careful, lest someone arrive inside out.
There’s an inverse relationship between the range of the transporter and the strength of its confinement beam. You can beam someone over a longer distance if there’s no/little interference, or pump up the confinement beam to punch through interference - but that reduces range. You can increase both range and confinement a bit by boosting power to the transporter as a whole, but that’s undesirable because it takes the system longer to channel that much energy safely - which means the passenger spends longer in the beam, which increases the risk of an accident. And of course, the transporter is a high-energy system anyway, and energy efficiency matters even on a starship.
Thing is, the interaction between these factors - range, confinement, and power requirements - is complicated enough that it can’t really be automated easily or safely. Thus, the transporter operator controls them manually, in response both to sensor readings and visual cues from the beam itself. Levers are used to provide tactile feedback, and allow the operator to control his console without looking at it - one of the rare instances on Starfleet vessels where it’s felt that the advantage of physical controls trumps the flexibility of touchscreens.
So there you have it - the three levers provide real-time control for transporter beam range, confinement field strength, and the amount of power feeding into the system. Now how’s that for a bit of Treknobabel? Totally out of my ass. 
Brake, throttle, clutch.
Contrast, Brightness, Hue
Hickory, Dickory, Dock
Absolutely beautiful.
If you’ve ever played with the T-Bar on a video switcher console, you know that it fades between two video feeds manually. That’s what the levers do. If the transporter operator was a dick, he could really mess around with you.
It controls the Heisenburg Compensator. How does the Heisenburg Compensator work? “It works very well, thank you.”