The logic of turning to the UN is straightforward: the U.S.-sponsored "peace process" - bilateral negotiations between Israel and the Palestinians under U.S. auspices - has failed, because a key premise of that process was that the U.S. government could bring the Israeli government to the table for a serious negotiation that would produce real Israeli compromise necessary for a solution. That premise has turned out to be spectacularly false.
The U.S. hasn't been able to bring the Israeli government to the table for a serious negotiation, not because it would be theoretically impossible to do so, but because "domestic political constraints" - the "Israel lobby" - have prevented the U.S. from exerting effective pressure on the Israeli government to move. Therefore, if the world wants to see resolution of the Israel-Palestine conflict anytime soon, it has to wrest control of the issue from Washington. And that's why moving the arena to the United Nations makes perfect sense.
Former Israeli peace negotiator Daniel Levy summed it up in the New York Times: "The U.S. cannot lead on an issue that it is so boxed in on by its domestic politics," Levy said. "And therefore, with the region in such rapid upheaval and the two-state solution dying, as long as the U.S. is paralyzed, others are going to have to step up."