I am controlling Ravenbloom Conservatory with 3 Might units, and my opponent attacks it with Anivia. Does Anivia's ability activate first to kill my units, or do I get to use the Battlefield ability first?

To determine the order of operations when Anivia attacks your battlefield containing a 3-Might unit and Ravenbloom Conservatory, we must look at the showdown timing rules.

The Ruling

Ravenbloom Conservatory’s ability will resolve first.

According to the showdown timing rules, when a unit moves to an occupied battlefield, the initial chain is populated by triggered abilities. Because the Ravenbloom Conservatory’s "When you defend here" ability is placed on the chain after Anivia’s "When I attack" ability, it resolves first due to the Last-In, First-Out (LIFO) nature of the chain.

Sequence of Events

1. Anivia moves to the battlefield: This initiates a showdown.
2. Triggers are placed on the chain:
   • Anivia's "When I attack" trigger is added to the chain.
   • Ravenbloom Conservatory's "When you defend here" trigger is added to the chain (on top of Anivia's).
3. Chain Resolution (LIFO - Last-In, First-Out):
   • The chain resolves in reverse order.
   • Ravenbloom Conservatory resolves first: You reveal the top card of your Main Deck. If it is a spell, you put it in your hand; otherwise, you recycle it.
   • Anivia's ability resolves second: Anivia deals 3 damage to all enemy units at the battlefield. Since your unit has 3 Might, it will be killed by this damage.

Summary

You resolve the Ravenbloom Conservatory ability first because it is the last trigger added to the chain, followed immediately by Anivia's ability, which deals the damage that kills your unit.

Note: I could not fully verify this ruling against the retrieved rules.