To be a bit more precise, the game loads another model and script, which is only ever used in that battle. An AI script makes the ATB bar locked at 254 (filled: 255) and progressively fills up the Limit bar (which was 0, since that 'character' was never used). When the limit bar reaches 255, the ATB advances one step, and it's Cloud's turn. It displays "LIMIT BREAK" like it does in any other battle, except without a name. During the time the limit level goes up, a camera script runs and some SFX are added in, too. Sephiroth's HP is set to 1, and his General Counter is set to end battle, as opposed to being his death counter, as you'd expect. This means you could potentially miss him and still win (although that's impossible, since I think Cloud's Hit% is 255%). If you wait long enough (Defend is not needed, you can simply wait), Sephiroth will attack, and Cloud will instantly counterattack, killing him. Sephiroth's attack takes 31/32 of the current HPs, making it impossible to lose this battle.
Some of this information may be inaccurate, and I may have forgotten a few things. Feel free to correct me.