Why does a rocket have far more payload capacity when placing an object in LEO compared to when it's placing the object in farther orbits, like on the Moon? If a rocket has sufficient thrust to take an heavier object into space, why wouldn't some additional stages of thrust allow it to take the same payload to the Moon?
The other answers are correct, but might be too hard to grasp intuitively. The simplest way to understand this is to reason the opposite way.
You have a rocket that can fly to the Moon. At some point in its flight, it already has enough speed to orbit the Earth, and some fuel to propel it to the Moon. If, instead of having extra fuel for the remainder of the journey, you'd put the same mass as payload, you'd have exactly this: more payload in LEO.
The short answer is: Tsiolkovsky rocket equation. You need some velocity to achieve some position (an orbit or a body) in space. Farther a position - more velocity. More velocity - more propellant mass, and this relation is not linear and not in favor of velocity.
$$\Delta v=v_e \ln(m_0/m_f)$$
$\Delta v$ - theoretical maximum increment of velocity,
$m_0$ - the initial mass, including tanks, engines, avionics, propellants and (of course) payload,
$m_f$ - the final mass, it can be payload only, depending of the rocket purpose and construction (payload can be very broad term, including a stage to fly to the Moon, Mars an so on, with it's own payload),
$v_e$ - the exhaust velocity of the selected type of the propellant for the selected type of the engine,
$\ln()$ - the natural logarithm.
If you add a stage, you add an initial mass, and yes, you can take the SAME payload to the Moon, but for the price of much heavier rocket. And it will be another rocket than ones for LEO. Or rocket may be the same but with less payload.
It perhaps become clearer when stating what rockets do. They change velocity. In space terms, that's delta-v.
A rocket stage can only change your velocity some limited amount. Different targets in space require different amounts of velocity change (Low orbit: 8km/s, low Moon orbit: 12km/s)
If your rocket stage can not give all the velocity change you need, a trick is needed. The trick is to replace some of the payload of the first rocket with another rocket, which can fire after the first one is spent.
The good news: You now have both the velocity change of the first rocket and the second rocket.
The bad news: Much of the payload is now the second rocket.
So in your example, when reaching low orbit, all the payload can be payload if you only intended to go that far. But if you need to go further to the Moon, extra velocity change is needed, so some of that payload must be an extra rocket.
Why is it easier to go halfway up a mountain than all the way to the top?
Assume that you are already in low Earth orbit. You have a payload X that you want to put in a higher orbit, say GEO. Then you can compute the deltaV needed to move that payload between orbits, and the fuel needed to create that deltaV.
But the fuel needed is not magically placed in LEO. You need to launch it (and probably the rocket stage that will use it) from the ground to LEO. That takes more fuel, which means you need either a bigger rocket, or several launches of smaller rockets.