How does quantum entanglement (AKA "spooky action at a distance") work in the concept that allows for the possibility of teleportation?
First, a bit about quantum entanglement. A pair of particles are said to be entangled if their quantum properties are correlated one with each other. For example, two electrons having the same quantum numbers n, l, ml (in other words exactly on the same energy level) have opposite spins (their spins are correlated). If somehow you can determine the spin of one electron you will know the spin of the other electron (because it is opposite to the the first one). If two particles are entangled the correlation between their properties will be valid even if there is a very huge distance between the particles. For example if you have two electrons with spins correlated, and one is on the earth and the other is on the moon, if you determine the spin of the electron on the earth as being s =+1/2, the spin of the electron on the moon will precisely be s =-1/2.
Now, in quantum mechanics to determine the quantum properties of a particle you need to make a measurement on them. This measurement itself is the one that "fixes" the properties of the particle. Until the measurement, you do not know exactly which values these quantum properties have.
Imagine you have a lot of these entangled pair of particles, and half of them you keep on earth and the other half you transport with a shuttle on the moon. Let also say that by using a certain procedure you make from the particles on the earth an object (like a frying pan for example) and then you measure the quantum state (you "fix" the quantum properties) of the all the particles in the frying pan you just made. Because the particles on the moon are entangled with the particles on the earth, they will suddenly assembly to make a second frying pan identical to the one on the earth. The process of measuring the quantum properties of the frying pan on the earth will destroy it forever, but at the same type a replica of it will be on the moon. Thus you will have succeeded to teleport the object on the moon.
Now, there is just an observation to be made here. The theory of relativity is saying that the interactions in universe are propagating with a maximum finite speed, and this speed is the speed of light. When you have entangled particles and you make a measurement of the properties on the first set of particles, exactly at the same moment the second set of particles will take the opposite quantum values of the properties. This means a contradiction with the theory of relativity and that in universe there can exist interaction speeds higher than the speed of light. This contradiction is not solved yet by physics.