Fire King monsters have their effects triggered when they're destroyed and sent to the Graveyard. Their boss monster, "Garunix", will destroy all other monsters on the field in the Standby Phase of the turn after it was destroyed by a card effect. This doesn't mean it needs to be on the field, so even cards like "Solemn Judgement" can triggers its effects. The other Fire King monsters can be Special Summoned from your hand when a Fire King is destroyed, keeping field advantage.
"Barong" is the main searcher of the deck. Not only it can add a Fire King monster, it can add any Fire King card, even "Fire King Fire Cycle" and "Assault of the Fire Kings". "Qilin" can send any FIRE monster from the Deck to the Graveyard, so they can be recycled with cards like "Fire King Fire Cycle" or "Call of the Haunted". "Yaksha" can destroy a card on your field or hand, so you can destroy a Fire King in your hand to get its effect as well. Another card that isn't a Fire King, but works in this deck as well, is "Sacred Phoenix of Nephthys", having a similar effect to "Garunix". It will destroy all Spell/Traps on the field in your Standby Phase.