If my opponent moves Vex Apathetic to an empty battlefield and I answer with Rengar Trophy Hunter, what units move back to base and does anyone stay at the battlefield?

To determine the outcome of this showdown, we must follow the sequence of events and the combat cleanup rules.

Sequence of Events

1. Opponent's Move: Your opponent moves Vex, Apathetic to an empty battlefield. This initiates a non-combat showdown and makes the battlefield Contested. Your opponent is the Attacker (Rule 442.1.a.1).
2. Your Response: You play Rengar, Trophy Hunter as a [Reaction] to the move. Because there is an enemy unit (Vex) at the battlefield, you may play him there. Upon playing Rengar, you become the Defender (Rule 442.1.a.2).
3. Vex's Trigger: Vex, Apathetic's ability triggers because you played a unit while she is at a battlefield. The ability ("When an opponent plays a unit while I'm at a battlefield, [Stun] it") is placed on the chain.
4. Chain Resolution: The chain resolves, and Rengar, Trophy Hunter becomes Stunned. He also gains the restriction that he cannot move for the rest of the turn.
5. Combat Begins: Once the chain clears and both players pass focus, the showdown transitions into a combat showdown.

Combat Resolution

• Combat Damage Step: Vex and Rengar exchange damage. Because Rengar is Stunned, he does not deal his Might in damage back to Vex (Rule 423.1.b).
• Combat Cleanup:
  1. Units heal.
  2. Recall: According to Rule 461.1.a.2 and FAQ #7302, attacking units are recalled to their controller's base if defenders are still present at the battlefield.
     • Since the Attacker (Vex) and the Defender (Rengar) are both present, the Attacker (Vex) is recalled to your opponent's base.
     • The Defender (Rengar) remains at the battlefield.

Final State

• Vex, Apathetic: Recalled to your opponent's base.
• Rengar, Trophy Hunter: Remains on the battlefield.
• Conquer: Because you are the only player with a unit remaining at the battlefield during the Resolution step, you Establish Control and Conquer the battlefield (Rule 461.5).

This interpretation is based on Rule 442.1.a (Attacker/Defender designations), Rule 461.1.a.2 (Combat Cleanup/Recall), Rule 423.1.b (Stun interaction), and RiftJudge FAQ #9936.

Note: I could not fully verify the specific damage values mentioned in the interaction as they were not provided in the rules context.