Let's start from the obvious things:
- When you bring the enemy pet below 50% health, your dragon indeed gets a 50% damage boost for one round.
- It's triggered on the backrow pets aswell. I don't know if the current movesets allow to do this but you can make use of it taking a wounded pet into the backrow and then performing an AoE attack.
- This racial puts dragons in specific relationships with flyers and beasts. The first ones can suffer badly because of that.
- From some reasons, if the enemy is still alive for several rounds, this racial triggers itself every second time as long as your dragon attacks and enemy's HP stay below 50%, so not only in the 1st round after the climax, but the 3rd and 5th aswell.
- This also causes a bug in the opposite way -- if you bring an enemy below 50% HP twice a row (e.g. Mana Surge killing two pets in two rounds), the second round doesn't grant you a bonus.
One can say that the second situation is needed because damage bonus itself causes increased probability to trigger this racial anyway. Still that makes the description kinda not worth the trust!