This question has been, by far, the trickiest one I've answered since I launched the website. It's also been the most interesting because it's really stretched my teaching ability to the limit. I've been thinking about how best to explain it for days and I think I've finally hit upon something which just about explains it, although I'm sure someone else could do it better. Firstly, let's talk about what shell theorem's second postulate is.
Imagine a perfectly spherical but hollow object. A ball-shell in other words. The wall of this shell has even thickness and the hollowed out centre is a perfect sphere also. Now imagine the shell has a strong gravitational pull because the shell-wall is thick enough. Essentially, we're imagining a perfectly hollowed out planet. The first postulate of the shell theorem is fairly simple. It says that if you're outside this shell, its gravity field will attract you from every angle and pull you in. You can imagine a point dead centre of the sphere which is what you're being pulled toward. This idea of a "centre of gravity" (the point through which weight appears to act) isn't too strange. Even though there is actually nothing in the centre of the sphere, you're still pulled as if there were. So far, so good. Where things get weird is with the second postulate.
The second postulate says that if you are inside the sphere, you float freely no matter where you are. If you're in the centre, you'd be equally attracted to the shell in all directions so you'd float. But the second postulate says that the same thing happens anywhere. Even if you're right over by one wall of the sphere, you still float freely and feel no attraction to the wall a few meters away from you. This doesn't seem right. After all, if you're right next to one side of the sphere and far away from the other side, shouldn't the force near you be stronger and ultimately win? Second postulate says no. How can that be right?
The reason this question has been so interesting is because all the conventional answers are just a bunch of equations and, as I've said before, equations are not explanations. If a students asks me what voltage is, I don't just write Voltage = Current x Resistance and say "there you go". Even though that's mathematically correct, this gives you no actual understanding. Equations are notations and language short-cuts for complicated ideas, but every equation is still describing something which could otherwise be expressed in words. The trick is to try and get to the bottom of what the equation really means and that's what I'm going to attempt to do. If I can't explain it in simple non-mathematical form, then I don't really understand it myself, I just "recognise the equation." Equations without explanations is lazy Science and lazy teaching. So here goes...
Let's imagine you are floating right over on the "east" side of the sphere (a sphere is non-directional and has no East, but I need to pick words to use). The east-most point is closest to you, so you feel its force very strongly. The western point of the sphere (the point most opposite to you) is far away, so it's effect is weak. But there's something else we need to factor in. There's a North and South point on the sphere too which are pulling you AWAY from the East point, half as strong as the West. These North and South points are half as far away, but there's two of them, so you have an equal force pulling you AWAY from East point as well as toward it.
And it doesn't stop there. Imagine the sphere having a North-East point. That point is also trying to pull you away from the East. And the same for the South-East point. In fact, and this is the crucial bit, there are more points on the inside of the sphere pulling you AWAY from East, and only one point on the East pulling you TOWARD it. These effects ultimately cancel out.
Imagine standing near the edge of a circle of people. The person nearest you is offering you £100 to go and shake their hand. But the two people either side of them are each offering you £99 to approach them instead. There's actually £198 worth of attraction trying to persuade you away from the £100 person. But you can't move toward either of them, because they're evenly matched - the amount is perfectly equal, so you're not actually attracted to either of them, you're attracted to both.
Inside the sphere, you end up with a balance of "lots of points attracting you weakly" verses "one point attracting you strongly" and these two effects cancel out perfectly. Because the inside of the is spherical, there are an infinite number of points trying to pull you away from anywhere you approach. Even when you're right beside one wall of the sphere, the infinte other walls are pulling you as well and they end up equalling the point directly next to you. As a result, you feel zero effect and you float there, freely.