It's actually not that they couldn't reach an agreement on the budget, per se. Gov. Christie demanded the state legislature agree to a "horse trade" and agree to two, unrelated demands in exchange for signing their budget. They agreed one but not the other, so Christie went to the beach. fwiw, his current approval rating is somewhere around 15%. He's done at the next election, and he knows it, so I suppose he feels free to do whatever he wants.
The holdup is that Gov. Christie wants to force a not-for-profit health insurance provider to divert money from its reserve fund into public health programs. The state legislature agreed to the deal and approved the governor's bill, but the Speaker has refused to put it for a vote, because he believes that the health insurance provider will simply shift the burden to its policy holders by raising its premiums. So aside from whether or not the Governor and/or Speaker are petulant brats, the stupid, stupid, stupid American health insurance system once again makes life hard for everybody.