What makes/breaks the hydrogen bonds between DNA and RNA during transcription?
I wouldn't really say RNA polymerase is "creating" the hydrogen bonds so much as it's thermodynamics that creates them. When we talk about an enzyme "creating" a bond, what we're generally referring to is an enzyme facilitating a reaction by lowering its activation energy so that it can proceed.

However, in the hydrogen bonding between base pairs, there's not really much activation energy to overcome. Before the formation, the nucleotide is just floating around in the cell (solvated by water), and forming a base pair is more energetically stable than not being paired. When a ribonucleotide happens to diffuse into contact with the unpaired DNA strand, the hydrogen bond will spontaneously form because it's energetically favorable. The thing about hydrogen bonds, though, is they're fairly weak, so there's an equilibrium between formation and breaking. When a mismatched nucleotide diffuses into that space, the hydrogen bonding isn't as strong, and it might be in an improper orientation, so it'll diffuse out again. When the correct nucleotide enters, the hydrogen bonds will form spontaneously, hold it there longer, and orientate it properly. This properly places the phosphates for backbone formation, which allows RNA polymerase to proceed with making the backbone.

As for the breaking of the hydrogen bonds, it's the energy released from the phosphodiester bond formation that causes the bonds to break at the other end of the RNA-DNA hybrid. To simplify things a lot, it's kind of like a zipper, where forward translocation of RNA polymerase along the DNA strand is driven by phosphodiester bond formation, which breaks the hydrogen bonds of the RNA-DNA hybrid somewhat mechanically, as the nascent RNA strand leaves through a different channel than the DNA does.

References: 

Structural Model of RNA Polymerase II Elongation Complex with Complete Transcription Bubble Reveals NTP Entry Route

Transcription elongation: Heterogeneous tracking of RNA polymerase and its biological implications