There's a few examples, let's use a power outage. So each time the player rolls to make the Pizza Party happen, but gets a 1-3, they tick a segment on the Interrupt clock. When the clock is full, in this example, the power goes out and puts a stall in the Tboys Pizza Party planning. It's up to the players how they want to respond to this, and it adds further story beats to respond to.
Given the way the dice work, the Interruption will very likely happen, and very likely before the Chill clock is filled. This is designed so that there's an interesting climax built it, but it could happen at any moment rather than being pre-planned.
Does that make sense?