Generating Bitcoin Public and Private Keys
Bitcoin uses point multiplication on the Elliptic Curve secp256k1 to generate a public key from a private key. Basically, this curve has a defined Generator point G, and a method for 'adding' two points together in a way to get a new point (EC Point Addition). 

Your private key is just a number, aka a Scalar, so to get your public key you just add the generator point to itself privkey number of times - basically scalar multiplication on the Elliptic curve. At the end, you'll end up with a point on the curve, with an X and Y coordinate, and this point is your public key. Those two coordinates of your point are what is referred to by the document you quoted, using 32 bytes for each coordinate plus a prefix.

The 0x04 prefix used is to denote the fact this is an uncompressed private key. You can also form a compressed version, by just using the X coordinate with either 0x02 or 0x03 prefix, because the Elliptic curve has only 2 values for Y coordinate for each X coordinate (one positive and one negative), so it's trivial to recover the Y coordinate if you know the X coordinate and the sign (based on your choice of prefix).