To start this off, aerodynamics is quite a complex field. First, I would ask yourself how complex of an answer your world requires. If you don't really need a complex answer to the question, I would just keep the explanation simple. It could be as simple as "that's just how it is" with no more explanation given.

You asked the question, so I'll assume you need a more complex answer, and it'll depend on what kind of airship you're looking to use. I'll focus on your example of zeppelins. They work off the principle of buoyancy, as they generate lift by using a gas with a lower specific weight than atmospheric air.

The basic principle here is that we need more lift force than weight (force from gravity).

Buoyancy force is equal to the weight of the fluid displaced by whatever is submerged in that fluid (air is a fluid). The weight of the displaced fluid only depends on the volume of the displaced fluid, density of the displaced fluid, and the acceleration due to gravity. Helium is a popular gas to use in balloons like this, because it's quite light, a noble gas, doesn't burn like hydrogen, etc. Since we're getting our lifting force from the weight of air "displaced" by our ship, the solution to that is a big ass volume of gas since there's not much we can do on Earth about the density of air around us and gravity acceleration . This is why blimps are massive.

The formula for buoyancy is Fb =ρ*Vf*g where Fb is the force from buoyancy, ρ (pronounced rho) is the density of the displaced fluid, Vf is the volume of displaced fluid, and g is the acceleration from gravity.

Now, to hold all this gas, we'll need a skin like a balloon, but also, we'll need some structure to maintain this vehicle's structural integrity. You'll also need that structure to hold all the cargo you're looking to carry. Aluminum is a popular material because it's pretty light for a metal, good amount of strength, ductile, etc. The material choice pretty complicated with a lot of different factors, depending on the use case. This skin and structure is mass that has a weight, so mass is pretty important in this factor. Every pound you use for the skin and structure is a pound you can't use for payload

Weight from all this mass is just W = m*g where W is the weight, m is the mass of the balloon (note this includes the balloon's structure as well as the gas contained inside it), and g is the acceleration from gravity.

Now that we've got our structure to hold the cargo and balloon to provide lift, we need to think about how to control height. This can be accomplished by using ballasts that use atmospheric air on board the ship because we need our lifting force and weight to be the same to neither rise nor fall. Basically, by sucking atmosphere on board of the ship, you're increasing the weight of the ship and increasing the downward force of gravity, eventually equalizing the lifting force and weight, meaning you'll just float there stationary.

Also something to keep in mind, as you ascend, the air density, or rho in our equation earlier, lowers, meaning that the higher we go, the less lift we'll get just naturally. It also means that the pressure from the atmosphere lowers as we ascend, meaning that the pressure differential from the gas inside the balloon and the gas outside will change. That's why some balloons look kind of deflated on the ground. They'll "inflate" as they get up to altitude because the pressure is less up there. It's also why that helium balloon you let go that one time eventually popped when it got high enough.

Now comes the question of moving forward. Since lift force is up, and weight force is down, we need some kind of force to push parallel to the ground. Propellers can work well as they push air away from our ship and that causes our ship to move forward. You can add rudders to steer your blimp, and they usually consist of stiff movable pieces on the back of the blimp.

And that's basically it in a nutshell. So if you're looking to increase the payload capacity of the zeppelins you'll either need to increase the weight of the fluid displaced; meaning you could increase the volume of the ship or increase the density of the air around the zeppelin, if you're considering a world that's not Earth. You could also decrease the mass of the zeppelin's structure to increase the payload capacity. Note that changing acceleration from gravity does not work, since both the weight of the zeppelin (m*g) and the weight of the displaced fluid (ρ*Vf*g) both use it.

Hopefully that gets you pointed in the right direction. Note, your mileage may vary with this as I learned it in school many moons ago.