To break the bank is to win so much that the casino, at least figuratively, runs out of money. Today it is used loosely for any big win, most often a jackpot on a slot, when players picture coins pouring out of the machine.
The phrase is older than slot machines. It originally described a player winning more than a gaming table’s cash reserve — its bank — so that play had to stop until the casino topped the table up. Modern casinos hold far larger reserves, so a literally broken bank is not a realistic scenario.
For operators, the idea still has a practical side. Large wins on progressive jackpots and high-limit tables must be covered by the casino’s bankroll, the jackpot’s funding and the game provider’s terms, and many operators celebrate big wins in the lobby when the player agrees.