professional mathematicians, please forgive.
A manifold is an n-dimensional object which, when you look at it on a small enough scale, can be considered in fewer than n dimensions. A good way of explaining this is an example: when you drive to the store, you don't have to care that the Earth is, in fact, a sphere. You just drive on a two-dimensional street map without having to know that you're curving around the center of the Earth. There isn't enough error on that scale to throw off your navigation by any noticeable amount. Therefore, we can consider a sphere (in this case, the Earth) to be a manifold.
A manifold is basically a way of having hidden structure which, from a small scale, can't be seen. From our perspective, the universe could be a manifold if it has hidden dimensions that we can't perceive but, on our small scale, we don't currently need them for things to work the way we expect. A four dimensional manifold is a way of describing our view in space and time of a universe with at least five dimensions.
Think of a shape with n faces (sides) and m vertices (corners). The dual of a shape is the figure that comes about by drawing a line from the middle of each face, to the middle of each adjacent face. The dual of your shape would be the one with m faces and n vertices - you're exchanging the number of vertices for the number of faces, and vice-versa. For example, a cube has eight vertices (eight corners) and six faces. The dual of a cube is the figure with six vertices and eight faces (an octahedron - two square pyramids with bottoms against each other). Wiki's got a good image of this:
Self-dual shapes are ones with the same number of vertices and faces. Since you get the dual by swapping the vertex number and the face number, it converts to an identical shape. A simple example of a self-dual shape is a tetrahedron or a square pyramid. In this case, the dual of the square pyramid is an upside-down square pyramid. For an easy to grasp concept, it's like one way you could describe a mirror universe. The math concept quoted in the post above is working with a similar idea but, instead of shapes, it's being performed on spaces.
For the purpose of explanation, imagine a set of numbers, each one corresponding to the location of one of the vertices of a pyramid. When grouped together, we can call this set of numbers a vector that describes the pyramid. The idea of being anti-self-dual is one where the dual is like taking the negative of this vector. It's like producing a similar shape, but with a different direction. (This is an oversimplification, but it helps get the idea across)
The principal bundle is something that exists only in mathematics - it doesn't really correspond to something you can see or touch. The important thing to know is that it's the name of the object whose dual is being taken. Think of it like a way of connecting point A to point B over a sphere.
If an operation is abelian, it means that the order of numbers on either side doesn't matter. Some great examples of this are multiplication and addition. It doesn't matter if you have 5x3 or 3x5, they both equal 15. The same works with addition: 7+8 = 8+7. Nonabelian is simply the case where this property doesn't happen - like subtraction and division.
A gauge theory is basically a special type of field where a particular thing holds true. In this case, think of a field as being the area in which a force can (and does) act, described by strength and direction. An example would be an electric or magnetic field. A field has something called a "Lagrangian", a function (a set of rules) that can be used to describe the field. For the field to be a gauge, the thing that must hold true is that Lagrangian function should stay the same when subjected to particular kinds of transformations. Thinking of these as transformations in the English sense of the word is OK to do.
A simple thought example of this is the electric field surrounding an electron. It doesn't matter whether you flip the electron upside down, turn it sideways, or inside out. It still has the same negative charge that repels other electrons out to a certain distance, regardless of how it points.
The end result of this is that an instanton is a type of theoretical thing that happens where all of these properties are true. I've really, really oversimplified it for the purposes of example, but the gist of it is that the manifold object is used to abstract the higher dimensions of our space-time into something we can understand, and the instanton is an idea representing a way of connecting multiple points under a particular set of circumstances that can occur.
This type of connection can bleed energy from something that appears to be a vacuum (the false vacuum). Since there's still energy there to be taken, it wasn't really a vacuum at all (defined as a lack of energy).
The suggestion is that the universe formed from a bubble that would have appeared to things inside it to be empty, but wasn't actually quite empty. The rest of it is just an explanation of one way that energy could have been lost.