Oh well. I tried to write a comment with a diagram but hit submit instead of preview :-(

Consider four directions on a plane. x axis (we'll call that |+>), y axis (we'll call that |->) y=-x (we'll call that |0>) and y=x (we'll call that |1>)

Modulo some constant factors, I hope it's obvious that you can build up some of those vectors from others:

|1> = |+> + |->

|-> = |0> + |1>

etc.

These are the directions of a plane polarized photon.

We setup some photons that are polarized in the |1> direction and then pass them through a polarization filter.

If the filter points along the |1> direction then all of them pass. If the filter passes along the |0> direction then none of them pass.

Now we put the filter along the |-> direction. What happens.

|1> = s|+> + s|-> (s is 1/sqrt(2) - which can be deduced from standard trig - the lines must be the same length)

When we measure along the |-> direction the s|-> part will pass the filter but the s|+> part wont.

But an individual photon can't get dimmer therefore it must either pass or not. Half the photons do pass and half don't (and it's random whether any one photon gets through the detector)

The ones that do get through are now in state |-> which is also |0>+|1> (again with factors of sqrt 2)

If we now measure along the |1> direction again we now lose half the photons again (due to that |0> component)

Quantum teleportation involves taking a photon in state a|0> + b|1> (for unknown values of a and b) and taking very careful measurements that don't destroy a and b but instead transfer them to another photon without us actually knowing what they are.