LEOlabs operates its own radar systems that point upwards-ish. The "beams" you see are the fields of view of their radars. When an object crosses that field of view going at orbital speeds, LEOlabs tracks it and uses the partial trajectory information to figure out the orbit of the object. From there, it can potentially associate that object with existing objects in its own and other databases (the US Space Force, which operates its own radars, is one of the best-known).

It then sells that information to spacecraft operators, who may be using the orbital information to determine if their spacecraft has a risk of hitting another object in space, or to figure out where their spacecraft are in the first place (usually when they're not talking to the ground).

Is there a big orbit registry somewhere? Yes. https://www.unoosa.org/oosa/osoindex/search-ng.jspx Each individual object is observed every few weeks, and then they estimate its current position with math and physics.

How do they track them? With big expensive radar systems, mostly paid for by the military. It's an international collaboration. https://en.wikipedia.org/wiki/Space_domain_awareness#Systems

LEOlabs operates and pays for their own radars. They're a commercial company.

I am pretty certain they must be dead reckoning the location with occasional updates.

That's how all space debris tracking works. Luckily orbits are rather predictable paths, but there's still atmospheric drag, lunar and solar gravity, non-uniform Earth gravity, solar photon pressure and measurement imprecision to deal with.

The US Air Force, for one, is responsible for tracking everything it sees. That generally means computing TLAs for each object. Whether they release this info, I don't know.

That capability was transferred to the Space Force recently. They make their data publicly available here: https://www.space-track.org/

Space Force.

https://en.wikipedia.org/wiki/18th_Space_Defense_Squadron