What carries the information for the Pauli exclusion principle to occur?

Looking for this kind of intuitive understanding of quantum mechanics is one of the things that makes QM hard to learn. You are thinking about how classical particles and waves behave, and are comparing that to quantum behavior. You look for similarities and they make sense because classical physics makes sense. But then the differences do not make sense.

It may be better to think of how quantum mechanical entities behave, and get used to the fact that they are not like classical entities. That is, rather than think about how an electron or photon is kind of like a particle and kind of like a wave, think about the new and different thing electron or photon is. What are its properties?

There are still profound conceptual difficulties. You have to accept that the universe behaves not just in ways that are different from what you expect, but contradict what you expect.

Suppose you had only ever heard of particles. You understand they have a mass, position, and velocity. They have a trajectory. Now someone introduces you to waves. They are different from particles. They don't have a point like position. They extend over a region, perhaps with fuzzy indefinite boundaries. They move, and yet don't have a trajectory. And yet they are something like particles. They carry energy and momentum. And yet they are different. They pass right through each other without bouncing off. It would be very confusing to think of a wave as some sort of particle. It makes more sense when you get used to what a wave is.

Here is a post that explains a photon is neither a particle nor a wave. How can a red light photon be different from a blue light photon? It shows how your intuition can be somewhat helpful and somewhat can get in the way. It turns out that electrons behave like this too, though classically we are more inclined to think of light as waves and electrons as particles.

Here is a link that begins to get at your question. Does the collapse of the wave function happen immediately everywhere? It gets more into how an electron is not like a particle nor a wave. A spread out electron can pass through two slits and interfere with itself on the other side. But then it can hit a single atom. It is very reasonable to wonder how information gets passed around. And yet this is not the right question to ask. There is no answer.

Historically, quantum mechanics addressed this with two sets of rules. One set of rules uses the Schrödinger equation to tell you how the electron's wave function changes with time. This tells you how the electron moves like a wave. These rules apply so long as the electron is not disturbed.

Then the electron gets disturbed, or "measured". Say it hits an atom. The wave function "collapses" to a new state. We don't see what happens during the collapse. We only see that we get a new state and we can describe how it changes with the Schrödinger equation. We can't predict what the new state will be from the old state. There may be a number of possible new states. We can predict probabilities of arriving at each one. Quantum mechanics has no mechanism that shows how information is passed around and no definite answers ahead of time to where it goes. This is called the Copenhagen interpretation of quantum mechanics.

This is something of a messy theory. It was accepted because it fit experiment very well. But it has problems. It isn't very clear exactly what a "measurement" is. It requires two sets of rules to describe what goes on, where one set would be more reasonable. It isn't deterministic.

Quantum mechanics is about a century old. Sometimes it takes a century or two to work out the kinks of a theory, and this is certainly true of quantum mechanics. About 50 years ago, the Everett or "Many Worlds" interpretation was proposed. It is now beginning to get acceptance. Here are a couple links that explain it. Parallel worlds probably exist. Here's why. And What is the Many Worlds interpretation? The jury is still out on this, but it is getting serious attention now.

The Many Worlds interpretation makes exactly the same experimentally testable predictions as the Copenhagen interpretation. But it eliminates the two sets of rules and questions of what a measurement is. It says the universe is deterministic. Where the Copenhagen interpretation says various outcomes might happen, the Many Worlds says all of the possible outcomes do happen every time. When they do, the universe splits into different worlds that can't talk to each other. The wave function doesn't collapse. The randomness comes in because we are split into many different versions of ourselves, and each version is only aware of one world.

This still doesn't provide a satisfactory, intuitive answer to your question. The wave function is all the information about a system that exists. Information gets passed around as described by the Schrödinger equation. Even though there may be two electrons, there is only one wave function that describes everything in all the many worlds. Two electrons never evolve so they have the same state because the Schrödinger equation doesn't permit it. This link explains why not. What causes Pauli's Exclusion Principle?

The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously

Let us talk simple, rock bottom quantum mechanics, not the high language of quantum field theory, which in any case is founded on rock bottom quantum mechanics.

By rock bottom I mean the postulates and the axiomatic postulated elementary particles, as in the standard model.

An important postulate is the wave function, a solution of a differential quantum mechanical equation that describes specific quantum numbers for each energy level of the solution in a potential. Lets take the simple hydrogen wavefunctions. There are three quantum numbers for each energy level, $n, l, m$. The Pauli exclusion fits the experimental observation that there are only single electrons at a single set of $n, l, m$. This makes sense in multi-electron atoms, as if there were no such limit, all electrons of the atom would hover in the lowest energy state, making chemistry impossible. (Even if a different law held, limiting the number in the same energy level chemistry would be different than the one we have.)

In a complex case of more than one atom the potential will not be that simple as to get analytic wavefunctions, but the principle of energy levels as the solution and quantum numbers assigned uniquely to each energy level again holds, thus the simple illustration with the single atoms holds.

If the energy levels are almost a continuum, as they are in the band theory of solids, there is no difficulty in fulfilling the Pauli principle and a large number of electrons.

You ask:

How does the information about the state of electrons get "passed around" so that other electrons in similar state can not have that same state? Is there some kind've information carrier?

The information is built in the wavefunction quantum numbers, no matter how complex the wavefunction is.

The fact that the two electrons in a given state have one spin up and the other down suggests that the exclusion of others is a result of magnetic dipole interactions.

Information does not get "passed around" because it is not even a material thing, it only exists for us and in our imagination which makes us believe that some relative positions of some objects in space have some meaning. But for the nature itself information is just not a thing.

Various particles of matter just keep on moving and interacting with each other until they happen to exchange their energy and more or less temporarily get into one of the locally stable states. How exactly they do the interactions deep down inside - I don't know, and I guess it is not even possible to know for sure, but today we have a way to estimate the shape of the stable regions of space and probabilities of going into ones or the others under various conditions.