The Royal Menagerie was added in patch 4.0 of the Stormblood expansion.
The Royal Menagerie becomes available at level 70 as part of the “Stormblood” quest, which is given by Pipin – The Lochs (X:32, Y:22.9) (Main Scenario).
The trial has a minimum ilevel of 280 to enter and rewards you with:
Allagan Tomestone of Verity: 10
Allagan Tomestone of Creation: 6