On History channels show The Universe they talked about solar storms "breaking" the magnetic field lines. This has confused me ever since. I thought magnetic field lines were mathematical representations, not a actual physical thing. How can a solar storm break the magnetic field?

You can't really see the magnetic field, but you can see the magnetic lines of force that the particles follow. In the same way you can see iron filings following the field of a bar magnet.

 

The reason for solar flares lies in the theory of magnetohydrodynamics and in particular it is believed that magnetic recombination is the mechanism. This is when the magnetic field lines of different magnetic domains (areas of uniform magnetisation) combine together and release energy.