a shot.

Here's the basics:

There are four fundamental forces in the universe. All interactions between things that we know of can be described as some combination of these four forces; the four forces are gravity, electromagnetism, the weak nuclear force, and the strong nuclear force. Of these four forces, electromagnetism is the one that influences your life the most visibly; gravity keeps you stuck to the face of the earth, but basically everything else you observe is the result of electromagnetic interactions.

For example: when you push on a door, the reason your hand doesn't just go through the door is because the electromagnetic force prevents the atoms in your hand from moving past the atoms in the door, and so the result is that the door has to move out of the way. Essentially all of the classical physics that you're used to (aside from gravity) are a result of this same phenomenon.

Of the four forces, three of them (electromagnetism and the strong and weak forces) are understood to operate by exchanging "force-carrying particles." The force-carrying particle for electromagnetism is the photon, which is the same particle that light is made of. (Light is another manifestation of electromagnetism--you can see because of the electromagnetic force.)

So now to your question: how to magnets work? Well, magnetism happens when the small-scale electromagnetic forces (that hold electrons in atoms and prevent larger structures from moving through each other) stack up to form a large-scale force.
So, from a certain way of looking at it, when two magnets repel each other, that's sort of a large-scale replication of the same thing that happens when the atoms in your hand repel the atoms in the door. And when two magnets attract each other, that's sort of a large-scale replication of when an electron is attracted to the protons in an atom and gets stuck in orbit around the atomic nucleus.

The mechanism that makes magnetism work can't be explained by classical mechanics (the physics that we're all used to dealing with in everyday life); it's the result of a quantum process, which is confusing to us because most quantum processes occur on such a small scale that we can't observe them directly, so we don't have a good intuition for understanding them. Magnetism completely defies explanation if you try to think about it in terms of things like Newton's laws of motion. Essentially, magnetism happens when two materials exchange photons in a certain way. (Remember, photons are the force-carrying particles for electromagnetism.) I don't understand the details of how this interaction works, so you'll have to look elsewhere for that, but that's the basic process that is going on; the magnet you're holding in your hand exchanges photons with the fridge, and the result of the exchange is that the magnet gets pulled towards the fridge.

I hope that helps. I also hope that someone who's actually an expert on this can come and show where my explanation is wrong.