PEG is the abbreviation for PolyEthylene Glycol. Just a chemical they add that makes the transformation work more efficiently.

DNA, as you probably know, is a double-stranded molecule, and the strands have complementary sequences. When the overlapping pieces are joined, one strand of one molecule overlaps with its complement on the other piece, and vice versa - the pieces aren't just joined end-to-end.

see http://en.wikipedia.org/wiki/Homologous_recombination for all the details.