I understand that aspect of it; I'm just confused as to how only having the daily key is enough to generate the identifiers. Wouldn't they also need the TimeIntervalNumber, according to the function?
According to the spec, the phone only generates a new identifier when the MAC address changes or on a new day. But since it's generated in accordance to a 10-minute time window, that means you'd try to derive their key with all 144 possible time windows for that day. And if you find one of those ID's in your list of contacts, then you know you were in contact with someone infected.