They're not saying the chance to burn goes up, they're saying as the match progresses the chance of it having scored at least one burn goes up.
Turn 1: One scald used, 30% chance a burn happened.
Turn 2: Two scalds used. if a burn happened first turn, then a burn happened, else a 30% roll on the 70% it didn't is added (21% added to running total, meaning a 51% chance a burn has happened).
Turn 3: Three scalds used. A 30% chance is rolled against the 49% chance a burn didn't happen in turn 1 or 2 combined. If true 14.7% is added to the 51% running total so there is now a 65.7% chance a burn has happened at least once in turn 1, 2 or 3.