Why can adding more polarization filters increase the amount of light that goes through them?

Indeed, it can be counterintuitive that adding a polarizing filter can increase the transmitted intensity, when each filter only 'removes' light.

Here's a slightly more intuitive way of thinking about it. A filter doesn't strictly remove light -- what it really does is add a light wave which destructively interferes with part of the incoming light. For example, a horizontal polarizing filter removes the horizontal part by transmitting an additional horizontally polarized wave out of phase by $180^\circ$. (This is true on a microscopic level, too. In the polarizing filter, electrons are driven by the incoming wave and, since they accelerate, they emit radiation of their own.)

Now let's say we add a diagonal polarizing filter in between horizontal and vertical filters. Thinking about the horizontal and vertical filters as "destroying light" in the usual way, you are correct in that none of the original light wave will make it through, regardless of what you put in between.

But the new diagonal filter adds a new diagonally polarized wave! It's part of this wave that makes it out.

There are two effect on a beam of light from passing through a polarizing filter.

• First the intensity is reduced as you write above. (For unpolarized light incident on a polarizer you average over all possible polarization and get $I = I_{up}/2$.)

• Second, the light that passes now has the polarization of the filter it just passed.

The second effect is what makes the trick work.

The same experiment can be demonstrated with a swinging rope. Place two slits behind each other (let space between them), the slits both in the same way oriented, take a rope through the slit, fix it at one end and swing the rope, of course in the direction of the slits. Now rotate the second slit. The rope no more will swing behind the second slit. But now if you place a third slit between the other two slits and this under 45°. The rope will swing all other his length and by this the swinging direction will be rotated to 90°.