Atoms have a nucleus and a bunch of electrons. When the atom receives energy, this energy is stored in the electrons or the nucleus. This isn't stable, however, and the atom always tries to release the energy back to the environment after a while. This energy is usually released as light, as a photon.

So the atom has a "base level" of energy which it is trying to stay at. "Ground state" means this state of lowest energy.

Electrons, protons and even neutrons all act like tiny loops of electric current, which produces a magnetic field. This is what we call spin. Because of the specific type of spin these particles have, they can only be found either up or down relative to any axis you decide to measure them on.

The nucleus, composed of protons and neutrons, has its own magnetic fields due to these spins. Since spins themselves are magnetic, the spin of the electrons "sense" the magnetic field due to the spin of the nucleus.

If you remember physics class, magnets want to align to the magnetic field applied to them. It's the same with spin: they prefer to align to the external magnetic field. Since electrons can only have up or down orientations along a given direction of magnetic field, the two orientations have slightly different energies.

When the spin of an electron flips from one direction to the other relative to the magnetic field of the nucleus, we have an hyperfine transition. Just as before, if the atom has more energy than the ground state, this energy can be stored in these spin directions. Once the energy is released, you get a photon of light with the same energy as the energy difference when the electron flipped.

This photon with that energy of the hyperfine transition has a very specific frequency, because energy and frequency are related.

A frequency tells you how often an event (say, an oscillation of the photon) happens in a given time interval. If you flip that, you get a period: how long it takes for one event to happen (T = 1/f).

Now, you count 9192631770 of these periods for the light emitted by a Casesium atom having those transitions. The time it took for you to count all those periods is what we call 1 second.

In short: We throw energy at a Caesium atom, it will be absorbed by the electrons. The electrons will flip their spins relative to magnetic field of the nucleus, storing part of that energy. When they flip back, they release that energy as light with a specific frequency. Find the period of that light, how long it takes that light to perform a full oscillation. Count 9192631770 periods of it, and the total time it took to count is 1 second.