From Livni's perspective this is a win-win situation. If Netanyahu capitulates, she could be seen both as supporting peace and as a likely coalition partner for Netanyahu in a unity government. Her statement is essentially a statement of intention to ally with Netanyahu in the event of a coalition breakdown. If the peace talks fail, Livni has the grounds to say she would have gone further for peace than Netanyahu (though its not clear she would actually make this claim in reality).
From Netanyahu's perspective, Livni's statement gives him more leverage with the far right. The specter of Likud breaking the coalition with right wing parties like Yisrael Beiteinu has always been a possibility, but this brings the possibility to the forefront. Netanyahu is unlikely to break ranks with the far right in the short-term, but he can use this statement to mitigate some of the damage a freeze extension would wreak on his coalition. He can leverage the threat of abandonment against threats from the right wing to cause a ruckus or break with Netanyahu.
For the United States, Livni's statements can be spun to indicate Israeli support for peace negotiations. The US can also use the statement to nudge Netanyahu to renew the settlement freeze (de facto or de jure).
Overall, Livni's statement is a positive development for the peace process because it injects stability into the calculus Netanyahu must make. And it strengthens the case that Netanyahu's best strategic option is a partial renewal of the settlement freeze.
*Discussion question: Did the US have a role, directly or indirectly in pushing Livni to make her statement?