Phase is cyclic, due to the periodicity of the sine function. When it is increasing and reaches 180 degrees it wraps around to -180 and continues with -179, -178 etc. When it is decreasing and reaches -180 degrees it wraps around to 180. The vertical lines show those wrap points. If you look in the graph controls you will see a button to "unwrap" the phase, which allows it to go to 181, 182 etc rather than wrapping, but you can end up with extremely large value ranges in unwrapped phase, partly because any time delays create a frequency-dependent phase shift that gets ever larger as frequency increases.
No, all that changes at the wrap points is the way the graph is drawn. Think of it more like reading the second hand of a clock, after 59 secs it starts again at 0 and counts up again, but time has not jumped or done anything other than progress as usual. The part of phase that gets plotted when wrapping is used is like showing only the seconds of a time.