The Rebirth Ability is a potential safety net from Game Over where the party has been Annihilated.
For it to occur, you must have Eiko in your party and she must have the summon "Phoenix". There is a small chance for the ability to engage, so you will be unlikely to force it. However, you can increase your chances by carrying 99 Phoenix Pinions, which can be synthesised at Master-Synths using Phoenix Down and Gyashal Greens.
The success rate is dependent on those Phoenix Pinions and the following formula; [(# of Phoenix Pinions / 256) * 100]. At 1 Phoenix Pinion, you have a 0.4% chance of it occurring, at 50 a 19.5% chance, and at 99 a 38.67% chance.